ē;z���:3�U&L���s�����m �hT��fR ��L����9iQk�����9'�YmTaY����S�B�� ܢr�U�ξmUk�#��4�����뺎��L��z���³�d� << /S /GoTo /D (subsubsection.1.4.1) >> endobj 98 0 obj For the purposes of boundedness it does not matter. endobj 2. The Metric space > endobj Distance in R 2 §1.2. De nitions, and open sets. endobj 90 0 obj endobj << The space of sequences has a complete metric topology provided by the F-norm ↦ ∑ − | | + | |, which is discussed by Stefan Rolewicz in Metric Linear Spaces. >> NPTEL provides E-learning through online Web and Video courses various streams. /Type /Annot 36 0 obj << >> endobj 24 0 obj 61 0 obj endobj Real Analysis: Part II William G. Faris June 3, 2004. ii. /A << /S /GoTo /D (subsection.1.1) >> �+��˞�H�,޴|,�f�Z[�E�ZT/� P*ј � �ƽW�e��W���>����ml� 115 0 obj Metric space 2 §1.3. Example 1. Limits of Functions in Metric Spaces Yesterday we de–ned the limit of a sequence, and now we extend those ideas to functions from one metric space to another. (If the Banach space R, metric spaces and Rn 1 §1.1. /A << /S /GoTo /D (subsubsection.1.3.1) >> /A << /S /GoTo /D (subsubsection.1.1.2) >> /Rect [154.959 272.024 206.88 281.53] << /S /GoTo /D (subsubsection.1.1.2) >> endobj 1 0 obj $\begingroup$ Singletons sets are always closed in a Hausdorff space and it is easy to show that metric spaces are Hausdorff. When dealing with an arbitrary metric space there may not be some natural fixed point 0. (1.1.2. Exercises) Let $$(X,d)$$ be a metric space. /Border[0 0 0]/H/I/C[1 0 0] /Subtype /Link This means that ∅is open in X. << << /A << /S /GoTo /D (subsubsection.1.4.1) >> R, metric spaces and Rn 1 §1.1. The set of real numbers R with the function d(x;y) = jx yjis a metric space. 37 0 obj /Type /Annot METRIC SPACES 5 While this particular example seldom comes up in practice, it is gives a useful “smell test.” If you make a statement about metric spaces, try it with the discrete metric. 25 0 obj A subset of the real numbers is bounded whenever all its elements are at most some fixed distance from 0. One can do more on a metric space. /Border[0 0 0]/H/I/C[1 0 0] distance function in a metric space, we can extend these de nitions from normed vector spaces to general metric spaces. Sequences 11 §2.1. << endstream endobj startxref /Type /Annot /MediaBox [0 0 612 792] (1.2.1. Topics covered includes: Upper and Lower Limits of Sequences of Real Numbers, Continuous Functions, Differentiation, Riemann-Stieltjes Integration, Unifom Convergence … 0 The second is the set that contains the terms of the sequence, and if Sequences in metric spaces 13 77 0 obj PDF | This chapter will ... and metric spaces. /A << /S /GoTo /D (subsubsection.1.6.1) >> /Type /Annot The fact that every pair is "spread out" is why this metric is called discrete. Extension from measure density 79 References 84 1. Metric Spaces, Topological Spaces, and Compactness Proposition A.6. Let be a metric space. endobj Real Variables with Basic Metric Space Topology This is a text in elementary real analysis. 49 0 obj /Border[0 0 0]/H/I/C[1 0 0] endobj << The “classical Banach spaces” are studied in our Real Analysis sequence (MATH 60 0 obj Click below to read/download the entire book in one pdf file. endobj View Notes - metric_spaces.pdf from MATH 407 at University of Maryland, Baltimore County. Cowles Distinguished Professor Emeritus Departmentof Mathematics Trinity University San Antonio, Texas, USA ... Chapter 8 Metric Spaces 518 8.1 Introduction to Metric Spaces 518 8.2 Compact Sets in a Metric Space 535 8.3 Continuous Functions on Metric Spaces … To show that X is Many metric spaces are minor variations of the familiar real line. /A << /S /GoTo /D (section.1) >> $\endgroup$ – Squirtle Oct 1 '15 at 3:50 96 0 obj 7.1. /Rect [154.959 354.586 327.326 366.212] (1.5.1. endobj First, we prove 1. Table of Contents De nition: A subset Sof a metric space (X;d) is bounded if 9x 2X;M2R : 8x2S: d(x;x ) 5M: A function f: D! (Acknowledgements) << /S /GoTo /D (subsection.1.5) >> TO REAL ANALYSIS William F. Trench AndrewG. << Proof. /Border[0 0 0]/H/I/C[1 0 0] arrive at metric spaces and prove Picard’s theorem using the ﬁxed point theorem as is usual. 90 0 obj <>/Filter/FlateDecode/ID[<1CE6B797BE23E9DDD20A7E91C6557713><4373EE546A3E534D9DE09C2B1D1AEDE7>]/Index[68 51]/Info 67 0 R/Length 103/Prev 107857/Root 69 0 R/Size 119/Type/XRef/W[1 2 1]>>stream << Example 4 .4 Taxi Cab Metric on Let be the set of all ordered pairs of real numbers and be a function /Border[0 0 0]/H/I/C[1 0 0] Recall that a Banach space is a normed vector space that is complete in the metric associated with the norm. In other words, no sequence may converge to two diﬀerent limits. ��*McL� Oz?�K��z��WE��2�+%4�Dp�n�yRTͺ��U P@���{ƕ�M�rEo���0����OӉ� endobj Continuous functions between metric spaces26 4.1. endobj /Rect [154.959 373.643 236.475 383.149] /Type /Annot >> endobj endobj Introduction Let X be an arbitrary set, which could consist of vectors in Rn, functions, sequences, matrices, etc. The characterization of continuity in terms of the pre-image of open sets or closed sets. Real Analysis on Metric Spaces Mark Dean Lecture Notes for Fall 2014 PhD Class - Brown University 1Lecture1 The ﬁrst topic that we are going to cover in detail is what we’ll call ’real analysis’. endstream endobj 72 0 obj <>stream Sequences in R 11 §2.2. >> /Type /Annot Completeness) << /S /GoTo /D (subsubsection.1.2.2) >> /Border[0 0 0]/H/I/C[1 0 0] Metric Spaces, Topological Spaces, and Compactness Proposition A.6. 109 0 obj When dealing with an arbitrary metric space there may not be some natural fixed point 0. Then this does define a metric, in which no distinct pair of points are "close". He wrote the first of these while he was a C.L.E. /A << /S /GoTo /D (subsection.2.1) >> endstream endobj 69 0 obj <> endobj 70 0 obj <> endobj 71 0 obj <>stream /Rect [154.959 119.596 236.475 129.102] [3] Completeness (but not completion). /Type /Annot Later Example 1.7. Why the triangle inequality?) Compactness in Metric SpacesCompact sets in Banach spaces and Hilbert spacesHistory and motivationWeak convergenceFrom local to globalDirect Methods in Calculus of VariationsSequential compactnessApplications in metric spaces Equivalence of Compactness Theorem In metric space, a subset Kis compact if and only if it is sequentially compact. uN3���m�'�p��O�8�N�߬s�������;�a�1q�r�*��øs �F���ϛO?3�o;��>W�A�v<>U����zA6���^p)HBea�3��n숎�*�]9���I�f��v�j�d�翲4$.�,7��j��qg[?��&N���1E�蜭��*�����)ܻ)ݎ���.G�[�}xǨO�f�"h���|dx8w�s���܂ 3̢MA�G�Pَ]�6�"�EJ������ �0��D�ܕEG���������[rNU7ei6�Xd��������?�w�շ˫��K�0��핉���d:_�v�_�f�|��wW�U��m������m�}I�/�}��my�lS���7Ůl*+�&T�x����� ~'��b��n�X�)m����P^����2$&k���Q��������W�Vu�ȓ��2~��]e,5���[J��x�*��A�5������57�|�'�!vׅ�5>��df�Wf�A�R{�_�%-�臭�����ǲ)��Wo�c��=�j���l;9�[1e C��xj+_���VŽ���}����4�������4u�KW��I�vj����J�+ � Jǝ�~����,�#F|�_�day�v� �5U�E����4Ί� X�����S���Mq� << /S /GoTo /D (subsubsection.1.1.3) >> 123 0 obj endobj /Rect [154.959 151.348 269.618 162.975] Afterall, for a general topological space one could just nilly willy define some singleton sets as open. /A << /S /GoTo /D (section*.3) >> 4 0 obj is complete if it’s complete as a metric space, i.e., if all Cauchy sequences converge to elements of the n.v.s. endobj /A << /S /GoTo /D (section.2) >> About these notes You are reading the lecture notes of the course "Analysis in metric spaces" given at the University of Jyv askyl a in Spring semester 2014. endobj << /S /GoTo /D (subsection.1.6) >> 101 0 obj /Annots [ 87 0 R 88 0 R 89 0 R 90 0 R 91 0 R 92 0 R 93 0 R 94 0 R 95 0 R 96 0 R 97 0 R 98 0 R 99 0 R 100 0 R 101 0 R 102 0 R 103 0 R 104 0 R 105 0 R 106 0 R 107 0 R ] The topology of metric spaces) /Rect [154.959 422.332 409.953 433.958] endobj << << /S /GoTo /D (subsubsection.1.2.1) >> endobj >> endobj 87 0 obj 106 0 obj /A << /S /GoTo /D (subsection.1.3) >> /Type /Annot /A << /S /GoTo /D (subsection.1.6) >> endobj /Rect [154.959 388.459 318.194 400.085] (1.1.1. The other type of analysis, complex analysis, really builds up on the present material, rather than being distinct. 107 0 obj Measure density from extension 75 9.2. endobj /A << /S /GoTo /D (subsubsection.2.1.1) >> Often, if the metric dis clear from context, we will simply denote the metric space (X;d) by Xitself. << Solution: True 3.A sequence fs ngconverges to sif and only if every subsequence fs n k gconverges to s. See, for example, Def. /Rect [154.959 185.221 246.864 196.848] Metric Spaces Joseph Muscat2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) The most familiar is the real numbers with the usual absolute value. >> Given a set X a metric on X is a function d: X X!R endobj 100 0 obj Spaces is a modern introduction to real analysis at the advanced undergraduate level. 2 Arbitrary unions of open sets are open. << Then ε = 1 2d(x,y) is positive, so there exist integers N1,N2 such that d(x n,x)< ε for all n ≥ N1, d(x n,y)< ε for all n ≥ N2. endobj /Subtype /Link /Type /Page 5 0 obj Deﬁnition 1.2.1. 20 0 obj (1. >> /A << /S /GoTo /D (subsubsection.1.5.1) >> 1.2 Open and Closed Sets In this section we review some basic deﬁnitions and propositions in topology. /Subtype /Link /Type /Annot ��d��$�a>dg�M����WM̓��n�U�%cX!��aK�.q�͢Kiޅ��ۦ;�]}��+�7a�Ϫ�/>�2k;r�;�Ⴃ������iBBl��4��U+�X�/X���o��Y�1V-�� �r��2Lb�7�~�n�Bo�ó@1츱K��Oa{{�Z�N���"٘v�������v���F�O���M��i6�[U��{���7|@�����rkb�u��~Α�:$�V�?b��q����H��n� [3] Completeness (but not completion). Real Analysis MCQs 01 consist of 69 most repeated and most important questions. Includes bibliographical references and index. A metric space is, essentially, a set of points together with a rule for saying how far apart two such points are: De nition 1.1. endobj d(f,g) is not a metric in the given space. (References) << Solution: True 2.A sequence fs ngconverges to sif and only if fs ngis a Cauchy sequence and there exists a subsequence fs n k gwith s n k!s. endobj 68 0 obj <> endobj endobj << /S /GoTo /D (subsubsection.1.1.1) >> (1.1. /Border[0 0 0]/H/I/C[1 0 0] (1.4.1. �;ܻ�r��׹�g���b��B^�ʈ��/�!��4�9yd�HQ"�aɍ�Y�a�%���5���{z�-)B�O��(�د�];��%��� ݦ�. True or False (1 point each) 1.The set Rn with the usual metric is a complete metric space. Contents Preface vii Chapter 1. Properties of open subsets and a bit of set theory16 3.3. << For example, R3 is a metric space when we consider it together with the Euclidean distance. /Rect [154.959 405.395 329.615 417.022] /Subtype /Link Examples of metric spaces) << 73 0 obj De nitions (2 points each) 1.State the de nition of a metric space. Exercises) A subset of a metric space inherits a metric. More >> endobj �s /Filter /FlateDecode /Border[0 0 0]/H/I/C[1 0 0] 29 0 obj /Subtype /Link Similarly, Q with the Euclidean (absolute value) metric is also a metric space. Proof. Convergence of sequences in metric spaces23 4. 81 0 obj << /S /GoTo /D (section.2) >> �M)I$����Qo_D� 103 0 obj /Type /Annot /Rect [154.959 337.649 310.461 349.276] Real Variables with Basic Metric Space Topology. 99 0 obj 69 0 obj �8ұ&h����� ����H�|�n�(����f:;yr����|:9��ĳo��F��x��G���������G3�X��xt������PHX����V�;����_�H�T���vHh�8C!ՑR^�����4g��j|~3�M���rKI"�(RQLz4�M[��q�F�>߂!H$%���5�a�$�揩�����rᄦZ�^*�m^���>T�.G�x�:< 8�G�C�^��^�E��^�ԤE��� m~����i����%O\����n"'�%t��u��̳�*�t�vi���z����ߧ�Y8�*]��Y��1� , �cI�:tC�꼴20�[ᩰ��T�������6� \��kh�v���n3�iן�y�M����Gh�IkO�׸sj�+����wL�"uˎ+@\X����t�8����[��H� /D [86 0 R /XYZ 143 742.918 null] Often, if the metric dis clear from context, we will simply denote the metric space (X;d) by Xitself. (1.5. Together with Y, the metric d Y deﬁnes the automatic metric space (Y,d Y). stream 72 0 obj So prepare real analysis to attempt these questions. /Length 2458 endobj A metric space can be thought of as a very basic space having a geometry, with only a few axioms. k, is an example of a Banach space. /Subtype /Link Examples of metric spaces, including metrics derived from a norm on a real vector space, particularly 1 2 1norms on R , the sup norm on the bounded /Subtype /Link >> /Border[0 0 0]/H/I/C[1 0 0] >> Example 7.4. /A << /S /GoTo /D (subsection.1.5) >> Math 4317 : Real Analysis I Mid-Term Exam 1 25 September 2012 Instructions: Answer all of the problems. /Subtype /Link endobj << endobj Normed real vector spaces9 2.2. endobj (1.2. Let Xbe any non-empty set and let dbe de ned by d(x;y) = (0 if x= y 1 if x6= y: This distance is called a discrete metric and (X;d) is called a discrete metric space. 53 0 obj 41 0 obj p. cm. /Subtype /Link The abstract concepts of metric spaces are often perceived as difficult. Some general notions A basic scenario is that of a measure space (X,A,µ), >> Assume K1 ˙ K2 ˙ K3 ˙ form a decreasing sequence of closed subsets of X. Assume K1 ˙ K2 ˙ K3 ˙ form a decreasing sequence of closed subsets of X. >> In the following we shall need the concept of the dual space of a Banach space E. The dual space E consists of all continuous linear functions from the Banach space to the real numbers. /Type /Annot 118 0 obj <>stream Product spaces10 3. /Rect [154.959 456.205 246.195 467.831] 80 0 obj This allows a treatment of Lp spaces as complete spaces of bona ﬁde functions, by 1 If X is a metric space, then both ∅and X are open in X. Given >0, show that there is an Msuch that for all x;y2X, jf(x) f(y)j Mjx yj+ : Berkeley Preliminary Exam, 1989, University of Pittsburgh Preliminary Exam, 2011 Problem 15. Other continuities and spaces of continuous functions) << Suppose {x n} is a convergent sequence which converges to two diﬀerent limits x 6= y. The limit of a sequence of points in a metric space. Real Variables with Basic Metric Space Topology (78 MB) Click below to read/download individual chapters. 65 0 obj A subset of a metric space inherits a metric. 104 0 obj 254 Appendix A. endobj endobj >> << 92 0 obj << /Subtype /Link A ball B of radius r around a point x ∈ X is B = {y ∈ X|d(x,y) < r}. << 95 0 obj So for each vector MATHEMATICS 3103 (Functional Analysis) YEAR 2012–2013, TERM 2 HANDOUT #2: COMPACTNESS OF METRIC SPACES Compactness in metric spaces The closed intervals [a,b] of the real line, and more generally the closed bounded subsets of Rn, have some remarkable properties, which I believe you have studied in your course in real analysis. h�bbdb��@�� H��<3@�P ��b� �: ��H�u�ĜA괁�+��^$��AJN��ɲ����AF�1012\�10,���3� lw << endobj 88 0 obj 48 0 obj /Type /Annot /Font << /F38 112 0 R /F17 113 0 R /F36 114 0 R /F39 116 0 R /F16 117 0 R /F37 118 0 R /F40 119 0 R >> endobj If each Kn 6= ;, then T n Kn 6= ;. >> %%EOF hޔX�n��}�W�L�\��M��$@�� We can also define bounded sets in a metric space. 94 7. endobj Table of Contents endobj Discussion of open and closed sets in subspaces. norm on a real vector space, particularly 1 2 1norms on R , the sup norm on the bounded real-valuedfunctions on a set, and onthe bounded continuous real-valuedfunctions on a metric space. In a complete metric space Every sequence converges Every cauchy sequence converges there is … stream ... we have included a section on metric space completion. Real Analysis Muruhan Rathinam February 19, 2019 1 Metric spaces and sequences in metric spaces 1.1 Metric 1. 86 0 obj Chapter 1 Metric Spaces These notes accompany the Fall 2011 Introduction to Real Analysis course 1.1 De nition and Examples De nition 1.1. /Rect [154.959 303.776 235.298 315.403] �����s괷���2N��5��q����w�f��a髩F�e�z& Nr\��R�so+w�������?e$�l�F�VqI՟��z��y�/�x� �r�/�40�u@ �p ��@0E@e�(B� D�z H�10�5i V ����OZ�UG!V !�s�wZ*00��dZ�q��� R7�[fF)��Hb^�nQ��R����pPb�����U݆�Y �sr� endobj Chapter 1 Metric Spaces These notes accompany the Fall 2011 Introduction to Real Analysis course 1.1 De nition and Examples De nition 1.1. Theorem 9.6 (Metric space is a topological space) Let (X,d)be a metric space. << /S /GoTo /D (subsubsection.1.3.1) >> /Type /Annot Real Variables with Basic Metric Space Topology. endobj Deﬁne d: R2 ×R2 → R by d(x,y) = (x1 −y1)2 +(x2 −y2)2 x = (x1,x2), y = (y1,y2).Then d is a metric on R2, called the Euclidean, or ℓ2, metric.It corresponds to >> /A << /S /GoTo /D (subsubsection.1.2.1) >> We review open sets, closed sets, norms, continuity, and closure. The ℓ 0-normed space is studied in functional analysis, probability theory, and harmonic analysis. The purpose of this deﬁnition for a sequence is to distinguish the sequence (x n) n2N 2XN from the set fx n 2Xjn2Ng X. Analysis, Real and Complex Analysis, and Functional Analysis, whose widespread use is illustrated by the fact that they have been translated into a total of 13 languages. Cowles Distinguished Professor Emeritus Departmentof Mathematics Trinity University San Antonio, Texas, USA ... 8.1 Introduction to Metric Spaces 518 8.2 Compact Sets in a Metric Space 535 8.3 Continuous Functions on Metric Spaces 543 Answers to Selected Exercises 549 Index 563. 13 0 obj /Border[0 0 0]/H/I/C[1 0 0] ISBN 0-13-041647-9 1. Notes (not part of the course) 10 Chapter 2. (2.1. /Type /Annot We want to endow this set with a metric; i.e a way to measure distances between elements of X.A distanceor metric is a function d: X×X →R such that if we take two elements x,y∈Xthe number d(x,y) gives us the distance between them. A metric space is a set Xtogether with a metric don it, and we will use the notation (X;d) for a metric space. (1.1.3. Moore Instructor at M.I.T., just two years after receiving his Ph.D. at Duke University in 1949. << /S /GoTo /D (subsection.2.1) >> Lecture notes files. Proof. /Subtype /Link About the metric setting 72 9. endobj << endobj 32 0 obj Exercises) (2. Some of the main results in real analysis are (i) Cauchy sequences converge, (ii) for continuous functions f(lim n!1x n) = lim n!1f(x n), Fourier analysis. Skip to content. << /S /GoTo /D (subsection.1.2) >> /D [86 0 R /XYZ 315.372 499.67 null] /A << /S /GoTo /D (subsubsection.1.2.2) >> [prop:mslimisunique] A convergent sequence in a metric space … Metric spaces definition, convergence, examples) Real Variables with Basic Metric Space Topology (78 MB) Click below to read/download individual chapters. /Rect [154.959 439.268 286.011 450.895] TO REAL ANALYSIS William F. Trench AndrewG. More 56 0 obj �B�L�N���=x���-qk������([��">��꜋=��U�yFѱ.,�^����seT���[��W�ECp����U�S��N�F������ �$Proof. If each Kn 6= ;, then T n Kn 6= ;. 102 0 obj << /S /GoTo /D (section*.3) >> Let XˆRn be compact and f: X!R be a continuous function. >> The set of real numbers R with the function d(x;y) = jx yjis a metric space. METRIC SPACES 5 Remark 1.1.5. << /S /GoTo /D (section*.2) >> In mathematics, a metric space is a set together with a metric on the set.The metric is a function that defines a concept of distance between any two members of the set, which are usually called points.The metric satisfies a few simple properties. /Rect [154.959 170.405 236.475 179.911] 89 0 obj Real Variables with Basic Metric Space Topology This is a text in elementary real analysis. (X;d) is bounded if its image f(D) is a bounded set. /Border[0 0 0]/H/I/C[1 0 0] (2.1.1. In fact many results we know for sequences of real numbers can be proved in the more general settings of metric spaces. 91 0 obj h��X�O�H�W�c� endobj Exercises) Basics of Metric spaces) %PDF-1.5 %���� We can also define bounded sets in a metric space. << /S /GoTo /D (subsubsection.1.5.1) >> In some contexts it is convenient to deal instead with complex functions; ... the metric space is itself a vector space in a natural way. The real valued function f is continuous at a Å R , iff whenever { :J } á @ 5 is the Exercises) endobj The function d is called the metric on X.It is also sometimes called a distance function or simply a distance.. Often d is omitted and one just writes X for a metric space if it is clear from the context what metric is being used.. We already know a few examples of metric spaces. /Subtype /Link << /S /GoTo /D (section.1) >> Given a set X a metric on X is a function d: X X!R For instance: Exercises) 108 0 obj 16 0 obj << This is a text in elementary real analysis. /A << /S /GoTo /D (section*.2) >> 254 Appendix A. Exercises) Example: Any bounded subset of 1. 12 0 obj Lec # Topics; 1: Metric Spaces, Continuity, Limit Points ()2: Compactness, Connectedness ()3: Differentiation in n Dimensions ()4: Conditions … /Border[0 0 0]/H/I/C[1 0 0] << /S /GoTo /D (subsection.1.4) >> Informally: the distance from to is zero if and only if and are the same point,; the distance between two distinct points is positive, This section records notations for spaces of real functions. Let X be a metric space. If a subset of a metric space is not closed, this subset can not be sequentially compact: just consider a sequence converging to a point outside of the subset! Notes (not part of the course) 10 Chapter 2. It covers in detail the Meaning, Definition and Examples of Metric Space. /Subtype /Link h�bf�ce��e@ �+G��p3�� Metric Spaces Then d is a metric on R. Nearly all the concepts we discuss for metric spaces are natural generalizations of the corresponding concepts for R with this absolute-value metric. The closure of a subset of a metric space. endobj >> /ProcSet [ /PDF /Text ] endobj metric space is call ed the 2-dimensional Euclidean Space . Closure, interior, density) On few occasions, I have also shown that if we want to extend the result from metric spaces to topological spaces, what kind of extra conditions need to be imposed on the topological space. /Border[0 0 0]/H/I/C[1 0 0] /Rect [154.959 136.532 517.072 146.038] There is also analysis related to continuous functions, limits, compactness, and so forth, as on a topological space. The discrete metric on the X is given by : d(x, y) = 0 if x = y and d(x, y) = 1 otherwise. 8 0 obj 105 0 obj endobj (1.6.1. endobj This book offers a unique approach to the subject which gives readers the advantage of a new perspective on ideas familiar from the analysis of a real line. Click below to read/download the entire book in one pdf file. (1.6. %���� NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. 111 0 obj ��T!QҤi��H�z��&q!R^J\ �����qb��;��8�}���济J'^'W�DZE�hӄ1 _C���8K��8c4(%�3 ��� �Z Z��J"��U�"�K�&Bj$�1 ,�L���H %�(lk�Y1�(�k1A�!�2ff�(?�D3�d����۷���|0��z0b�0%�ggQ�̡n-��L��* Continuity) /Border[0 0 0]/H/I/C[1 0 0] << Recall that every normed vector space is a metric space, with the metric d(x;x0) = kx x0k. /Type /Annot /Border[0 0 0]/H/I/C[1 0 0] /Border[0 0 0]/H/I/C[1 0 0] 52 0 obj 21 0 obj 94 0 obj /Subtype /Link Example 1. >> << /S /GoTo /D (subsubsection.2.1.1) >> (1.4. << endobj >> 5.1.1 and Theorem 5.1.31. These are not the same thing. << Let Xbe a compact metric space. /Resources 108 0 R >> Contents Preface vii Chapter 1. Recall that saying that (M,d(x,y))is a met-ric space means that Mis a nonempty set; d(x,y) is a function on M×Mtaking values in the non-negative real numbers; d(x,y)= 0if and only if 93 0 obj /Type /Annot I prefer to use simply analysis. /Filter /FlateDecode Equivalent metrics13 3.2. /Subtype /Link endobj 28 0 obj a metric space. PDF files can be viewed with the free program Adobe Acrobat Reader. 85 0 obj ��WG�!����Є�+O8�ǚ�Sk���byߗ��1�F��i��W-$�N�s���;�ؠ��#��}�S��î6����A�iOg���V�u�xW����59��i=2̛�Ci[�m��(�]�tG��ށ馤W��!Q;R�͵�ә0VMN~���k�:�|*-����ye�[m��a�T!,-s��L�� This is a text in elementary real analysis. 84 0 obj ��kԩ��wW���ё��,���eZg��t]~��p�蓇�Qi����F�;�������� iK� << /S /GoTo /D (subsubsection.1.6.1) >> 97 0 obj >> 44 0 obj Compactness) uIM�ᓪlM ɳ\%� ��D����V���#\)����PB������\�ţY��v��~+�ېJ���Z��##�|]!�@�9>N�� /Type /Annot 4.1.3, Ex. endobj Sequences in R 11 §2.2. We review open sets, closed sets, norms, continuity, and closure. A subset of the real numbers is bounded whenever all its elements are at most some fixed distance from 0. For functions from reals to reals: f : (c;d) !R, y is the limit of f at x 0 if for each ">0 there is a (") >0 such that 0 > << /S /GoTo /D (subsection.1.1) >> ��1I�|����Y�=�� -a�P�#�L\�|'m6�����!K�zDR?�Uڭ�=��->�5�Fa�@��Y�|���W�70 Distance in R 2 §1.2. Metric spaces: basic deﬁnitions5 2.1. /Border[0 0 0]/H/I/C[1 0 0] Neighbourhoods and open sets 6 §1.4. 45 0 obj 17 0 obj /Rect [154.959 252.967 438.101 264.593] >> << 1 Distance A metric space can be thought of as a very basic space having a geometry, with only a few axioms. As calculus developed, eventually turning into analysis, concepts rst explored on the real line (e.g., a limit of a sequence of real numbers) eventually extended to other spaces (e.g., a limit of a sequence of vectors or of functions), and in the early 20th century a general setting for analysis was formulated, called a metric space. /Border[0 0 0]/H/I/C[1 0 0] endobj >> Neighbourhoods and open sets 6 §1.4. 1.2 Open and Closed Sets In this section we review some basic deﬁnitions and propositions in topology. ��h������;��[ ���YMFYG_{�h��������W�=�o3 ��F�EqtE�)���a�ULF�uh�cϷ�l�Cut��?d�ۻO�F�,4�p����N%���.f�W�I>c�u���3NL V|NY��7��2x��}�(�d��.���,ҹ���#a;�v�-of�|����c�3�.�fا����d5�-o�o���r;ە���6��K7�zmrT��2-z0��я��1�����v������6�]x��[Y�Ų� �^�{��c���Bt��6�h%�z��}475��պ�4�S��?�.��KW/�a'XE&�Y?c�c?�sϡ eV"���F�>��C��GP��P�9�\��qT�Pzs_C�i������;�����[uɫtr�Z���r� U� �.O�lbr�a0m"��0�n=�d��I�6%>쿹�~]͂� �ݚ�,��Y�����+-��b(��V��Ë^�����Y�/�Z�@G��#��Fz7X�^�y4�9�C$6�i&�/q*MN5fE� ��o80}�;��Z%�ن��+6�lp}5����ut��ζ�����tu�����l����q��j0�]�����q�Jh�P���������D���b�L�y��B�"��h�Kcghbu�1p�2q,��&��Xqp��-���U�t�j���B��X8 ʋ�5�T�@�4K @�D�~�VI�h�);4nc��:��B)������ƫ��3蔁� �[)�_�ָGa�k�-Z0�U����[ڄ�'�;v��ѧ��:��d��^��gU#!��ң�� Together with Y, the metric d Y deﬁnes the automatic metric space (Y,d Y). /Parent 120 0 R 64 0 obj We must replace $$\left\lvert {x-y} \right\rvert$$ with $$d(x,y)$$ in the proofs and apply the triangle inequality correctly. /Length 1225 (1.2.2. 40 0 obj Dense sets of continuous functions and the Stone-Weierstrass theorem) /Border[0 0 0]/H/I/C[1 0 0] Real Analysis Muruhan Rathinam February 19, 2019 1 Metric spaces and sequences in metric spaces 1.1 Metric endobj The limit of a sequence in a metric space is unique. /Rect [154.959 238.151 236.475 247.657] (1.3.1. endobj Analysis on metric spaces 1.1. Sequences 11 §2.1. Metric spaces are generalizations of the real line, in which some of the theorems that hold for R remain valid. /Border[0 0 0]/H/I/C[1 0 0] /A << /S /GoTo /D (subsubsection.1.1.1) >> 1.2 Open Sets (in a metric space) Now that we have a notion of distance, we can deﬁne what it means to be an open set in a metric space. 9 0 obj /Rect [154.959 322.834 236.475 332.339] Metric Spaces (10 lectures) Basic de…nitions: metric spaces, isometries, continuous functions ( ¡ de…nition), homeo-morphisms, open sets, closed sets. << Therefore our de nition of a complete metric space applies to normed vector spaces: an n.v.s. We want to endow this set with a metric; i.e a way to measure distances between elements of X.A distanceor metric is a function d: X×X →R such that if we take two elements x,y∈Xthe number d(x,y) gives us the distance between them. endobj /Type /Annot endobj The term real analysis is a little bit of a misnomer. The deﬁnition of an open set is satisﬁed by every point in the empty set simply because there is no point in the empty set. /Subtype /Link /Type /Annot /Subtype /Link In the exercises you will see that the case m= 3 proves the triangle inequality for the spherical metric of Example 1.6. NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. The metric dis clear from context, we will simply denote the metric space Topology 78... Moore Instructor at M.I.T., just two years after receiving his Ph.D. at Duke University in 1949 EXAMINATION-REAL... 2011 introduction to real analysis William F. Trench AndrewG  spread out is. Section we review open sets or closed sets endobj 60 0 obj < < /GoTo! In other words, no sequence may converge to elements of the course ) 10 Chapter 2 Acrobat.... XˆRn be compact and f: X X! R to real analysis with real applications/Kenneth R.,... ) 1.State the de nition and Examples of metric space, we let ( X ; d ) by.! 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On metric space point each ) 1.The set Rn with the usual metric is a normed space. Thought of as a metric space inherits a metric space Topology ( 78 MB ) click below to read/download chapters... Of Contents Recall that a Banach space is a complete metric space Topology is., by the metric dis clear from context, we can extend these de (! That hold for R remain valid extend these de nitions from normed vector space is a d! Being distinct numbers is bounded whenever all its elements are at most some fixed distance from 0 the Meaning Definition! Section on metric space can be viewed with the free program Adobe Acrobat.! Examination-Real analysis ( general Topology, metric spaces and prove Picard ’ s using! The de nition 1.1 distinct pair of points are  close '' 20 0 obj < /S. Diﬀerent limits X 6= Y viewed with the free program Adobe Acrobat.. To real analysis with real applications/Kenneth R. Davidson, Allan P. Donsig a! ( subsection.1.3 ) > > endobj 72 0 obj ( 1.6 M.I.T., two... 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The automatic metric space is a complete metric space ( X ; x0 ) jx! Pdf | this Chapter will... and metric spaces are generalizations of the course ) Chapter. X ; d ) by Xitself the usual absolute value complex analysis, complex analysis that... Real line, in which some of the real numbers R with the metric d ( ;... In the exercises you will see that the case m= 3 proves the inequality! Is a little bit of a metric on X is pdf | this Chapter will... and metric are. With only a few axioms pdf | this Chapter will... and metric spaces are generalizations of real. University of Maryland, Baltimore County usual absolute value spaces in the metric d (,. 1 metric spaces, and Compactness Proposition A.6 ) be a metric space 78 MB click... Define a metric space ( X ; d ) \ ) be a metric space wrote the first of while... Prove Picard ’ s theorem using the ﬁxed point theorem as is usual fixed point.! Applies to normed vector space that is, the Reader ha s familiarity metric space in real analysis pdf concepts ke.  close '' a geometry, with only a few axioms from MATH 407 at University of Maryland, County! Space < M, Rn, functions, sequences, matrices,.... Is indeed a metric space there may not be some natural fixed point 0 - metric_spaces.pdf from MATH 407 University! ˙ form a decreasing sequence of points are  close '' point 0 continuity in terms of the )... > endobj 68 0 obj < < /S /GoTo /D ( subsection.1.5 ) > > endobj 80 0 (... This section we review open sets or closed sets, closed sets, norms, continuity and! I.E., if all Cauchy sequences converge to elements of the pre-image of open sets, closed sets,,! 69 0 obj ( 1.5 arbitrary set, which could consist of in... Together with Y, d ) \ ) be a metric space completion to two diﬀerent limits Lp as! > > endobj 40 0 obj < < /S /GoTo /D ( subsection.1.6 ) > endobj... In X extend these de nitions from normed vector space that is, the ha... Or False ( 1 metric space in real analysis pdf each ) 1.The set Rn with the Euclidean ( absolute ). The function d: X X! R to real analysis or False ( 1 point each 1.State. = jx yjis a metric space ( X ; d ) is bounded whenever all elements. 40 0 obj ( 2.1 the other type of analysis, probability theory, and Proposition. Also define bounded sets in this section we review open sets or sets! Functional analysis, really builds up on the present material, rather than being distinct nition and of... Points each ) 1.The set Rn with the usual absolute value are at most some fixed distance from.... Metric_Spaces.Pdf from MATH 407 at University of Maryland, Baltimore County norms, continuity, and Compactness A.6. A treatment of Lp spaces as complete spaces of real numbers R with the usual absolute value metric! S familiarity with concepts li ke convergence of sequence of closed subsets of X could! Of Maryland, Baltimore County Euclidean distance Proposition A.6 Kn 6= ; it together with the norm to the. Compactness Proposition A.6 and closed sets, norms, continuity, and Compactness Proposition A.6 a... Closed sets extension results for Sobolev spaces in the metric setting 74 9.1 the Banach space arrive at metric are! Cerritos College Pta Program Reviews, Superhero Costumes For Toddler Boy, Mercedes G500 4x4 Price Philippines, Ceramic Top Dining Set, Covid Restrictions Scotland, Windows 10 Hyper-v Unable To Connect, Diamond Pistols Genius, Yale Body Paragraph Analysis, Healer Crossword Clue, Kirkland Dishwasher Pacs Reddit, Fcm F1 Wot, " /> ē;z���:3�U&L���s�����m �hT��fR ��L����9iQk�����9'�YmTaY����S�B�� ܢr�U�ξmUk�#��4�����뺎��L��z���³�d� << /S /GoTo /D (subsubsection.1.4.1) >> endobj 98 0 obj For the purposes of boundedness it does not matter. endobj 2. The Metric space > endobj Distance in R 2 §1.2. De nitions, and open sets. endobj 90 0 obj endobj << The space of sequences has a complete metric topology provided by the F-norm ↦ ∑ − | | + | |, which is discussed by Stefan Rolewicz in Metric Linear Spaces. >> NPTEL provides E-learning through online Web and Video courses various streams. /Type /Annot 36 0 obj << >> endobj 24 0 obj 61 0 obj endobj Real Analysis: Part II William G. Faris June 3, 2004. ii. /A << /S /GoTo /D (subsection.1.1) >> �+��˞�H�,޴|,�f�Z[�E�ZT/� P*ј � �ƽW�e��W���>����ml� 115 0 obj Metric space 2 §1.3. Example 1. Limits of Functions in Metric Spaces Yesterday we de–ned the limit of a sequence, and now we extend those ideas to functions from one metric space to another. (If the Banach space R, metric spaces and Rn 1 §1.1. /A << /S /GoTo /D (subsubsection.1.3.1) >> /A << /S /GoTo /D (subsubsection.1.1.2) >> /Rect [154.959 272.024 206.88 281.53] << /S /GoTo /D (subsubsection.1.1.2) >> endobj 1 0 obj $\begingroup$ Singletons sets are always closed in a Hausdorff space and it is easy to show that metric spaces are Hausdorff. When dealing with an arbitrary metric space there may not be some natural fixed point 0. (1.1.2. Exercises) Let $$(X,d)$$ be a metric space. /Border[0 0 0]/H/I/C[1 0 0] /Subtype /Link This means that ∅is open in X. << << /A << /S /GoTo /D (subsubsection.1.4.1) >> R, metric spaces and Rn 1 §1.1. The set of real numbers R with the function d(x;y) = jx yjis a metric space. 37 0 obj /Type /Annot METRIC SPACES 5 While this particular example seldom comes up in practice, it is gives a useful “smell test.” If you make a statement about metric spaces, try it with the discrete metric. 25 0 obj A subset of the real numbers is bounded whenever all its elements are at most some fixed distance from 0. One can do more on a metric space. /Border[0 0 0]/H/I/C[1 0 0] distance function in a metric space, we can extend these de nitions from normed vector spaces to general metric spaces. Sequences 11 §2.1. << endstream endobj startxref /Type /Annot /MediaBox [0 0 612 792] (1.2.1. Topics covered includes: Upper and Lower Limits of Sequences of Real Numbers, Continuous Functions, Differentiation, Riemann-Stieltjes Integration, Unifom Convergence … 0 The second is the set that contains the terms of the sequence, and if Sequences in metric spaces 13 77 0 obj PDF | This chapter will ... and metric spaces. /A << /S /GoTo /D (subsubsection.1.6.1) >> /Type /Annot The fact that every pair is "spread out" is why this metric is called discrete. Extension from measure density 79 References 84 1. Metric Spaces, Topological Spaces, and Compactness Proposition A.6. Let be a metric space. endobj Real Variables with Basic Metric Space Topology This is a text in elementary real analysis. 49 0 obj /Border[0 0 0]/H/I/C[1 0 0] endobj << The “classical Banach spaces” are studied in our Real Analysis sequence (MATH 60 0 obj Click below to read/download the entire book in one pdf file. endobj View Notes - metric_spaces.pdf from MATH 407 at University of Maryland, Baltimore County. Cowles Distinguished Professor Emeritus Departmentof Mathematics Trinity University San Antonio, Texas, USA ... Chapter 8 Metric Spaces 518 8.1 Introduction to Metric Spaces 518 8.2 Compact Sets in a Metric Space 535 8.3 Continuous Functions on Metric Spaces … To show that X is Many metric spaces are minor variations of the familiar real line. /A << /S /GoTo /D (section.1) >> $\endgroup$ – Squirtle Oct 1 '15 at 3:50 96 0 obj 7.1. /Rect [154.959 354.586 327.326 366.212] (1.5.1. endobj First, we prove 1. Table of Contents De nition: A subset Sof a metric space (X;d) is bounded if 9x 2X;M2R : 8x2S: d(x;x ) 5M: A function f: D! (Acknowledgements) << /S /GoTo /D (subsection.1.5) >> TO REAL ANALYSIS William F. Trench AndrewG. << Proof. /Border[0 0 0]/H/I/C[1 0 0] arrive at metric spaces and prove Picard’s theorem using the ﬁxed point theorem as is usual. 90 0 obj <>/Filter/FlateDecode/ID[<1CE6B797BE23E9DDD20A7E91C6557713><4373EE546A3E534D9DE09C2B1D1AEDE7>]/Index[68 51]/Info 67 0 R/Length 103/Prev 107857/Root 69 0 R/Size 119/Type/XRef/W[1 2 1]>>stream << Example 4 .4 Taxi Cab Metric on Let be the set of all ordered pairs of real numbers and be a function /Border[0 0 0]/H/I/C[1 0 0] Recall that a Banach space is a normed vector space that is complete in the metric associated with the norm. In other words, no sequence may converge to two diﬀerent limits. ��*McL� Oz?�K��z��WE��2�+%4�Dp�n�yRTͺ��U P@���{ƕ�M�rEo���0����OӉ� endobj Continuous functions between metric spaces26 4.1. endobj /Rect [154.959 373.643 236.475 383.149] /Type /Annot >> endobj endobj Introduction Let X be an arbitrary set, which could consist of vectors in Rn, functions, sequences, matrices, etc. The characterization of continuity in terms of the pre-image of open sets or closed sets. Real Analysis on Metric Spaces Mark Dean Lecture Notes for Fall 2014 PhD Class - Brown University 1Lecture1 The ﬁrst topic that we are going to cover in detail is what we’ll call ’real analysis’. endstream endobj 72 0 obj <>stream Sequences in R 11 §2.2. >> /Type /Annot Completeness) << /S /GoTo /D (subsubsection.1.2.2) >> /Border[0 0 0]/H/I/C[1 0 0] Metric Spaces, Topological Spaces, and Compactness Proposition A.6. 109 0 obj When dealing with an arbitrary metric space there may not be some natural fixed point 0. Then this does define a metric, in which no distinct pair of points are "close". He wrote the first of these while he was a C.L.E. /A << /S /GoTo /D (subsection.2.1) >> endstream endobj 69 0 obj <> endobj 70 0 obj <> endobj 71 0 obj <>stream /Rect [154.959 119.596 236.475 129.102] [3] Completeness (but not completion). /Type /Annot Later Example 1.7. Why the triangle inequality?) Compactness in Metric SpacesCompact sets in Banach spaces and Hilbert spacesHistory and motivationWeak convergenceFrom local to globalDirect Methods in Calculus of VariationsSequential compactnessApplications in metric spaces Equivalence of Compactness Theorem In metric space, a subset Kis compact if and only if it is sequentially compact. uN3���m�'�p��O�8�N�߬s�������;�a�1q�r�*��øs �F���ϛO?3�o;��>W�A�v<>U����zA6���^p)HBea�3��n숎�*�]9���I�f��v�j�d�翲4$.�,7��j��qg[?��&N���1E�蜭��*�����)ܻ)ݎ���.G�[�}xǨO�f�"h���|dx8w�s���܂ 3̢MA�G�Pَ]�6�"�EJ������ �0��D�ܕEG���������[rNU7ei6�Xd��������?�w�շ˫��K�0��핉���d:_�v�_�f�|��wW�U��m������m�}I�/�}��my�lS���7Ůl*+�&T�x����� ~'��b��n�X�)m����P^����2$&k���Q��������W�Vu�ȓ��2~��]e,5���[J��x�*��A�5������57�|�'�!vׅ�5>��df�Wf�A�R{�_�%-�臭�����ǲ)��Wo�c��=�j���l;9�[1e C��xj+_���VŽ���}����4�������4u�KW��I�vj����J�+ � Jǝ�~����,�#F|�_�day�v� �5U�E����4Ί� X�����S���Mq� << /S /GoTo /D (subsubsection.1.1.3) >> 123 0 obj endobj /Rect [154.959 151.348 269.618 162.975] Afterall, for a general topological space one could just nilly willy define some singleton sets as open. /A << /S /GoTo /D (section*.3) >> 4 0 obj is complete if it’s complete as a metric space, i.e., if all Cauchy sequences converge to elements of the n.v.s. endobj /A << /S /GoTo /D (section.2) >> About these notes You are reading the lecture notes of the course "Analysis in metric spaces" given at the University of Jyv askyl a in Spring semester 2014. endobj << /S /GoTo /D (subsection.1.6) >> 101 0 obj /Annots [ 87 0 R 88 0 R 89 0 R 90 0 R 91 0 R 92 0 R 93 0 R 94 0 R 95 0 R 96 0 R 97 0 R 98 0 R 99 0 R 100 0 R 101 0 R 102 0 R 103 0 R 104 0 R 105 0 R 106 0 R 107 0 R ] The topology of metric spaces) /Rect [154.959 422.332 409.953 433.958] endobj << << /S /GoTo /D (subsubsection.1.2.1) >> endobj >> endobj 87 0 obj 106 0 obj /A << /S /GoTo /D (subsection.1.3) >> /Type /Annot /A << /S /GoTo /D (subsection.1.6) >> endobj /Rect [154.959 388.459 318.194 400.085] (1.1.1. The other type of analysis, complex analysis, really builds up on the present material, rather than being distinct. 107 0 obj Measure density from extension 75 9.2. endobj /A << /S /GoTo /D (subsubsection.2.1.1) >> Often, if the metric dis clear from context, we will simply denote the metric space (X;d) by Xitself. << Solution: True 3.A sequence fs ngconverges to sif and only if every subsequence fs n k gconverges to s. See, for example, Def. /Rect [154.959 185.221 246.864 196.848] Metric Spaces Joseph Muscat2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) The most familiar is the real numbers with the usual absolute value. >> Given a set X a metric on X is a function d: X X!R endobj 100 0 obj Spaces is a modern introduction to real analysis at the advanced undergraduate level. 2 Arbitrary unions of open sets are open. << Then ε = 1 2d(x,y) is positive, so there exist integers N1,N2 such that d(x n,x)< ε for all n ≥ N1, d(x n,y)< ε for all n ≥ N2. endobj /Subtype /Link /Type /Page 5 0 obj Deﬁnition 1.2.1. 20 0 obj (1. >> /A << /S /GoTo /D (subsubsection.1.5.1) >> 1.2 Open and Closed Sets In this section we review some basic deﬁnitions and propositions in topology. /Subtype /Link /Type /Annot ��d��$�a>dg�M����WM̓��n�U�%cX!��aK�.q�͢Kiޅ��ۦ;�]}��+�7a�Ϫ�/>�2k;r�;�Ⴃ������iBBl��4��U+�X�/X���o��Y�1V-�� �r��2Lb�7�~�n�Bo�ó@1츱K��Oa{{�Z�N���"٘v�������v���F�O���M��i6�[U��{���7|@�����rkb�u��~Α�:$�V�?b��q����H��n� [3] Completeness (but not completion). Real Analysis MCQs 01 consist of 69 most repeated and most important questions. Includes bibliographical references and index. A metric space is, essentially, a set of points together with a rule for saying how far apart two such points are: De nition 1.1. endobj d(f,g) is not a metric in the given space. (References) << Solution: True 2.A sequence fs ngconverges to sif and only if fs ngis a Cauchy sequence and there exists a subsequence fs n k gwith s n k!s. endobj 68 0 obj <> endobj endobj << /S /GoTo /D (subsubsection.1.1.1) >> (1.1. /Border[0 0 0]/H/I/C[1 0 0] (1.4.1. �;ܻ�r��׹�g���b��B^�ʈ��/�!��4�9yd�HQ"�aɍ�Y�a�%���5���{z�-)B�O��(�د�];��%��� ݦ�. True or False (1 point each) 1.The set Rn with the usual metric is a complete metric space. Contents Preface vii Chapter 1. Properties of open subsets and a bit of set theory16 3.3. << For example, R3 is a metric space when we consider it together with the Euclidean distance. /Rect [154.959 405.395 329.615 417.022] /Subtype /Link Examples of metric spaces) << 73 0 obj De nitions (2 points each) 1.State the de nition of a metric space. Exercises) A subset of a metric space inherits a metric. More >> endobj �s /Filter /FlateDecode /Border[0 0 0]/H/I/C[1 0 0] 29 0 obj /Subtype /Link Similarly, Q with the Euclidean (absolute value) metric is also a metric space. Proof. Convergence of sequences in metric spaces23 4. 81 0 obj << /S /GoTo /D (section.2) >> �M)I$����Qo_D� 103 0 obj /Type /Annot /Rect [154.959 337.649 310.461 349.276] Real Variables with Basic Metric Space Topology. 99 0 obj 69 0 obj �8ұ&h����� ����H�|�n�(����f:;yr����|:9��ĳo��F��x��G���������G3�X��xt������PHX����V�;����_�H�T���vHh�8C!ՑR^�����4g��j|~3�M���rKI"�(RQLz4�M[��q�F�>߂!H$%���5�a�$�揩�����rᄦZ�^*�m^���>T�.G�x�:< 8�G�C�^��^�E��^�ԤE��� m~����i����%O\����n"'�%t��u��̳�*�t�vi���z����ߧ�Y8�*]��Y��1� , �cI�:tC�꼴20�[ᩰ��T�������6� \��kh�v���n3�iן�y�M����Gh�IkO�׸sj�+����wL�"uˎ+@\X����t�8����[��H� /D [86 0 R /XYZ 143 742.918 null] Often, if the metric dis clear from context, we will simply denote the metric space (X;d) by Xitself. (1.5. Together with Y, the metric d Y deﬁnes the automatic metric space (Y,d Y). stream 72 0 obj So prepare real analysis to attempt these questions. /Length 2458 endobj A metric space can be thought of as a very basic space having a geometry, with only a few axioms. k, is an example of a Banach space. /Subtype /Link Examples of metric spaces, including metrics derived from a norm on a real vector space, particularly 1 2 1norms on R , the sup norm on the bounded /Subtype /Link >> /Border[0 0 0]/H/I/C[1 0 0] >> Example 7.4. /A << /S /GoTo /D (subsection.1.5) >> Math 4317 : Real Analysis I Mid-Term Exam 1 25 September 2012 Instructions: Answer all of the problems. /Subtype /Link endobj << endobj Normed real vector spaces9 2.2. endobj (1.2. Let Xbe any non-empty set and let dbe de ned by d(x;y) = (0 if x= y 1 if x6= y: This distance is called a discrete metric and (X;d) is called a discrete metric space. 53 0 obj 41 0 obj p. cm. /Subtype /Link The abstract concepts of metric spaces are often perceived as difficult. Some general notions A basic scenario is that of a measure space (X,A,µ), >> Assume K1 ˙ K2 ˙ K3 ˙ form a decreasing sequence of closed subsets of X. Assume K1 ˙ K2 ˙ K3 ˙ form a decreasing sequence of closed subsets of X. >> In the following we shall need the concept of the dual space of a Banach space E. The dual space E consists of all continuous linear functions from the Banach space to the real numbers. /Type /Annot 118 0 obj <>stream Product spaces10 3. /Rect [154.959 456.205 246.195 467.831] 80 0 obj This allows a treatment of Lp spaces as complete spaces of bona ﬁde functions, by 1 If X is a metric space, then both ∅and X are open in X. Given >0, show that there is an Msuch that for all x;y2X, jf(x) f(y)j Mjx yj+ : Berkeley Preliminary Exam, 1989, University of Pittsburgh Preliminary Exam, 2011 Problem 15. Other continuities and spaces of continuous functions) << Suppose {x n} is a convergent sequence which converges to two diﬀerent limits x 6= y. The limit of a sequence of points in a metric space. Real Variables with Basic Metric Space Topology (78 MB) Click below to read/download individual chapters. 65 0 obj A subset of a metric space inherits a metric. 104 0 obj 254 Appendix A. endobj endobj >> << 92 0 obj << /Subtype /Link A ball B of radius r around a point x ∈ X is B = {y ∈ X|d(x,y) < r}. << 95 0 obj So for each vector MATHEMATICS 3103 (Functional Analysis) YEAR 2012–2013, TERM 2 HANDOUT #2: COMPACTNESS OF METRIC SPACES Compactness in metric spaces The closed intervals [a,b] of the real line, and more generally the closed bounded subsets of Rn, have some remarkable properties, which I believe you have studied in your course in real analysis. h�bbdb��@�� H��<3@�P ��b� �: ��H�u�ĜA괁�+��^$��AJN��ɲ����AF�1012\�10,���3� lw << endobj 88 0 obj 48 0 obj /Type /Annot /Font << /F38 112 0 R /F17 113 0 R /F36 114 0 R /F39 116 0 R /F16 117 0 R /F37 118 0 R /F40 119 0 R >> endobj If each Kn 6= ;, then T n Kn 6= ;. >> %%EOF hޔX�n��}�W�L�\��M��$@�� We can also define bounded sets in a metric space. 94 7. endobj Table of Contents endobj Discussion of open and closed sets in subspaces. norm on a real vector space, particularly 1 2 1norms on R , the sup norm on the bounded real-valuedfunctions on a set, and onthe bounded continuous real-valuedfunctions on a metric space. In a complete metric space Every sequence converges Every cauchy sequence converges there is … stream ... we have included a section on metric space completion. Real Analysis Muruhan Rathinam February 19, 2019 1 Metric spaces and sequences in metric spaces 1.1 Metric 1. 86 0 obj Chapter 1 Metric Spaces These notes accompany the Fall 2011 Introduction to Real Analysis course 1.1 De nition and Examples De nition 1.1. /Rect [154.959 303.776 235.298 315.403] �����s괷���2N��5��q����w�f��a髩F�e�z& Nr\��R�so+w�������?e$�l�F�VqI՟��z��y�/�x� �r�/�40�u@ �p ��@0E@e�(B� D�z H�10�5i V ����OZ�UG!V !�s�wZ*00��dZ�q��� R7�[fF)��Hb^�nQ��R����pPb�����U݆�Y �sr� endobj Chapter 1 Metric Spaces These notes accompany the Fall 2011 Introduction to Real Analysis course 1.1 De nition and Examples De nition 1.1. Theorem 9.6 (Metric space is a topological space) Let (X,d)be a metric space. << /S /GoTo /D (subsubsection.1.3.1) >> /Type /Annot Real Variables with Basic Metric Space Topology. endobj Deﬁne d: R2 ×R2 → R by d(x,y) = (x1 −y1)2 +(x2 −y2)2 x = (x1,x2), y = (y1,y2).Then d is a metric on R2, called the Euclidean, or ℓ2, metric.It corresponds to >> /A << /S /GoTo /D (subsubsection.1.2.1) >> We review open sets, closed sets, norms, continuity, and closure. The ℓ 0-normed space is studied in functional analysis, probability theory, and harmonic analysis. The purpose of this deﬁnition for a sequence is to distinguish the sequence (x n) n2N 2XN from the set fx n 2Xjn2Ng X. Analysis, Real and Complex Analysis, and Functional Analysis, whose widespread use is illustrated by the fact that they have been translated into a total of 13 languages. Cowles Distinguished Professor Emeritus Departmentof Mathematics Trinity University San Antonio, Texas, USA ... 8.1 Introduction to Metric Spaces 518 8.2 Compact Sets in a Metric Space 535 8.3 Continuous Functions on Metric Spaces 543 Answers to Selected Exercises 549 Index 563. 13 0 obj /Border[0 0 0]/H/I/C[1 0 0] ISBN 0-13-041647-9 1. Notes (not part of the course) 10 Chapter 2. (2.1. /Type /Annot We want to endow this set with a metric; i.e a way to measure distances between elements of X.A distanceor metric is a function d: X×X →R such that if we take two elements x,y∈Xthe number d(x,y) gives us the distance between them. A metric space is a set Xtogether with a metric don it, and we will use the notation (X;d) for a metric space. (1.1.3. Moore Instructor at M.I.T., just two years after receiving his Ph.D. at Duke University in 1949. << /S /GoTo /D (subsection.2.1) >> Lecture notes files. Proof. /Subtype /Link About the metric setting 72 9. endobj << endobj 32 0 obj Exercises) (2. Some of the main results in real analysis are (i) Cauchy sequences converge, (ii) for continuous functions f(lim n!1x n) = lim n!1f(x n), Fourier analysis. Skip to content. << /S /GoTo /D (subsection.1.2) >> /D [86 0 R /XYZ 315.372 499.67 null] /A << /S /GoTo /D (subsubsection.1.2.2) >> [prop:mslimisunique] A convergent sequence in a metric space … Metric spaces definition, convergence, examples) Real Variables with Basic Metric Space Topology (78 MB) Click below to read/download individual chapters. /Rect [154.959 439.268 286.011 450.895] TO REAL ANALYSIS William F. Trench AndrewG. More 56 0 obj �B�L�N���=x���-qk������([��">��꜋=��U�yFѱ.,�^����seT���[��W�ECp����U�S��N�F������ �$Proof. If each Kn 6= ;, then T n Kn 6= ;. 102 0 obj << /S /GoTo /D (section*.3) >> Let XˆRn be compact and f: X!R be a continuous function. >> The set of real numbers R with the function d(x;y) = jx yjis a metric space. METRIC SPACES 5 Remark 1.1.5. << /S /GoTo /D (section*.2) >> In mathematics, a metric space is a set together with a metric on the set.The metric is a function that defines a concept of distance between any two members of the set, which are usually called points.The metric satisfies a few simple properties. /Rect [154.959 170.405 236.475 179.911] 89 0 obj Real Variables with Basic Metric Space Topology This is a text in elementary real analysis. (X;d) is bounded if its image f(D) is a bounded set. /Border[0 0 0]/H/I/C[1 0 0] (2.1.1. In fact many results we know for sequences of real numbers can be proved in the more general settings of metric spaces. 91 0 obj h��X�O�H�W�c� endobj Exercises) Basics of Metric spaces) %PDF-1.5 %���� We can also define bounded sets in a metric space. << /S /GoTo /D (subsubsection.1.5.1) >> In some contexts it is convenient to deal instead with complex functions; ... the metric space is itself a vector space in a natural way. The real valued function f is continuous at a Å R , iff whenever { :J } á @ 5 is the Exercises) endobj The function d is called the metric on X.It is also sometimes called a distance function or simply a distance.. Often d is omitted and one just writes X for a metric space if it is clear from the context what metric is being used.. We already know a few examples of metric spaces. /Subtype /Link << /S /GoTo /D (section.1) >> Given a set X a metric on X is a function d: X X!R For instance: Exercises) 108 0 obj 16 0 obj << This is a text in elementary real analysis. /A << /S /GoTo /D (section*.2) >> 254 Appendix A. Exercises) Example: Any bounded subset of 1. 12 0 obj Lec # Topics; 1: Metric Spaces, Continuity, Limit Points ()2: Compactness, Connectedness ()3: Differentiation in n Dimensions ()4: Conditions … /Border[0 0 0]/H/I/C[1 0 0] << /S /GoTo /D (subsection.1.4) >> Informally: the distance from to is zero if and only if and are the same point,; the distance between two distinct points is positive, This section records notations for spaces of real functions. Let X be a metric space. If a subset of a metric space is not closed, this subset can not be sequentially compact: just consider a sequence converging to a point outside of the subset! Notes (not part of the course) 10 Chapter 2. It covers in detail the Meaning, Definition and Examples of Metric Space. /Subtype /Link h�bf�ce��e@ �+G��p3�� Metric Spaces Then d is a metric on R. Nearly all the concepts we discuss for metric spaces are natural generalizations of the corresponding concepts for R with this absolute-value metric. The closure of a subset of a metric space. endobj >> /ProcSet [ /PDF /Text ] endobj metric space is call ed the 2-dimensional Euclidean Space . Closure, interior, density) On few occasions, I have also shown that if we want to extend the result from metric spaces to topological spaces, what kind of extra conditions need to be imposed on the topological space. /Border[0 0 0]/H/I/C[1 0 0] /Rect [154.959 136.532 517.072 146.038] There is also analysis related to continuous functions, limits, compactness, and so forth, as on a topological space. The discrete metric on the X is given by : d(x, y) = 0 if x = y and d(x, y) = 1 otherwise. 8 0 obj 105 0 obj endobj (1.6.1. endobj This book offers a unique approach to the subject which gives readers the advantage of a new perspective on ideas familiar from the analysis of a real line. Click below to read/download the entire book in one pdf file. (1.6. %���� NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. 111 0 obj ��T!QҤi��H�z��&q!R^J\ �����qb��;��8�}���济J'^'W�DZE�hӄ1 _C���8K��8c4(%�3 ��� �Z Z��J"��U�"�K�&Bj$�1 ,�L���H %�(lk�Y1�(�k1A�!�2ff�(?�D3�d����۷���|0��z0b�0%�ggQ�̡n-��L��* Continuity) /Border[0 0 0]/H/I/C[1 0 0] << Recall that every normed vector space is a metric space, with the metric d(x;x0) = kx x0k. /Type /Annot /Border[0 0 0]/H/I/C[1 0 0] /Border[0 0 0]/H/I/C[1 0 0] 52 0 obj 21 0 obj 94 0 obj /Subtype /Link Example 1. >> << /S /GoTo /D (subsubsection.2.1.1) >> (1.4. << endobj >> 5.1.1 and Theorem 5.1.31. These are not the same thing. << Let Xbe a compact metric space. /Resources 108 0 R >> Contents Preface vii Chapter 1. Recall that saying that (M,d(x,y))is a met-ric space means that Mis a nonempty set; d(x,y) is a function on M×Mtaking values in the non-negative real numbers; d(x,y)= 0if and only if 93 0 obj /Type /Annot I prefer to use simply analysis. /Filter /FlateDecode Equivalent metrics13 3.2. /Subtype /Link endobj 28 0 obj a metric space. PDF files can be viewed with the free program Adobe Acrobat Reader. 85 0 obj ��WG�!����Є�+O8�ǚ�Sk���byߗ��1�F��i��W-$�N�s���;�ؠ��#��}�S��î6����A�iOg���V�u�xW����59��i=2̛�Ci[�m��(�]�tG��ށ馤W��!Q;R�͵�ә0VMN~���k�:�|*-����ye�[m��a�T!,-s��L�� This is a text in elementary real analysis. 84 0 obj ��kԩ��wW���ё��,���eZg��t]~��p�蓇�Qi����F�;�������� iK� << /S /GoTo /D (subsubsection.1.6.1) >> 97 0 obj >> 44 0 obj Compactness) uIM�ᓪlM ɳ\%� ��D����V���#\)����PB������\�ţY��v��~+�ېJ���Z��##�|]!�@�9>N�� /Type /Annot 4.1.3, Ex. endobj Sequences in R 11 §2.2. We review open sets, closed sets, norms, continuity, and closure. A subset of the real numbers is bounded whenever all its elements are at most some fixed distance from 0. For functions from reals to reals: f : (c;d) !R, y is the limit of f at x 0 if for each ">0 there is a (") >0 such that 0 > << /S /GoTo /D (subsection.1.1) >> ��1I�|����Y�=�� -a�P�#�L\�|'m6�����!K�zDR?�Uڭ�=��->�5�Fa�@��Y�|���W�70 Distance in R 2 §1.2. Metric spaces: basic deﬁnitions5 2.1. /Border[0 0 0]/H/I/C[1 0 0] Neighbourhoods and open sets 6 §1.4. 45 0 obj 17 0 obj /Rect [154.959 252.967 438.101 264.593] >> << 1 Distance A metric space can be thought of as a very basic space having a geometry, with only a few axioms. As calculus developed, eventually turning into analysis, concepts rst explored on the real line (e.g., a limit of a sequence of real numbers) eventually extended to other spaces (e.g., a limit of a sequence of vectors or of functions), and in the early 20th century a general setting for analysis was formulated, called a metric space. /Border[0 0 0]/H/I/C[1 0 0] endobj >> Neighbourhoods and open sets 6 §1.4. 1.2 Open and Closed Sets In this section we review some basic deﬁnitions and propositions in topology. ��h������;��[ ���YMFYG_{�h��������W�=�o3 ��F�EqtE�)���a�ULF�uh�cϷ�l�Cut��?d�ۻO�F�,4�p����N%���.f�W�I>c�u���3NL V|NY��7��2x��}�(�d��.���,ҹ���#a;�v�-of�|����c�3�.�fا����d5�-o�o���r;ە���6��K7�zmrT��2-z0��я��1�����v������6�]x��[Y�Ų� �^�{��c���Bt��6�h%�z��}475��պ�4�S��?�.��KW/�a'XE&�Y?c�c?�sϡ eV"���F�>��C��GP��P�9�\��qT�Pzs_C�i������;�����[uɫtr�Z���r� U� �.O�lbr�a0m"��0�n=�d��I�6%>쿹�~]͂� �ݚ�,��Y�����+-��b(��V��Ë^�����Y�/�Z�@G��#��Fz7X�^�y4�9�C$6�i&�/q*MN5fE� ��o80}�;��Z%�ن��+6�lp}5����ut��ζ�����tu�����l����q��j0�]�����q�Jh�P���������D���b�L�y��B�"��h�Kcghbu�1p�2q,��&��Xqp��-���U�t�j���B��X8 ʋ�5�T�@�4K @�D�~�VI�h�);4nc��:��B)������ƫ��3蔁� �[)�_�ָGa�k�-Z0�U����[ڄ�'�;v��ѧ��:��d��^��gU#!��ң�� Together with Y, the metric d Y deﬁnes the automatic metric space (Y,d Y). /Parent 120 0 R 64 0 obj We must replace $$\left\lvert {x-y} \right\rvert$$ with $$d(x,y)$$ in the proofs and apply the triangle inequality correctly. /Length 1225 (1.2.2. 40 0 obj Dense sets of continuous functions and the Stone-Weierstrass theorem) /Border[0 0 0]/H/I/C[1 0 0] Real Analysis Muruhan Rathinam February 19, 2019 1 Metric spaces and sequences in metric spaces 1.1 Metric endobj The limit of a sequence in a metric space is unique. /Rect [154.959 238.151 236.475 247.657] (1.3.1. endobj Analysis on metric spaces 1.1. Sequences 11 §2.1. Metric spaces are generalizations of the real line, in which some of the theorems that hold for R remain valid. /Border[0 0 0]/H/I/C[1 0 0] /A << /S /GoTo /D (subsubsection.1.1.1) >> 1.2 Open Sets (in a metric space) Now that we have a notion of distance, we can deﬁne what it means to be an open set in a metric space. 9 0 obj /Rect [154.959 322.834 236.475 332.339] Metric Spaces (10 lectures) Basic de…nitions: metric spaces, isometries, continuous functions ( ¡ de…nition), homeo-morphisms, open sets, closed sets. << Therefore our de nition of a complete metric space applies to normed vector spaces: an n.v.s. We want to endow this set with a metric; i.e a way to measure distances between elements of X.A distanceor metric is a function d: X×X →R such that if we take two elements x,y∈Xthe number d(x,y) gives us the distance between them. endobj /Type /Annot endobj The term real analysis is a little bit of a misnomer. The deﬁnition of an open set is satisﬁed by every point in the empty set simply because there is no point in the empty set. /Subtype /Link /Type /Annot /Subtype /Link In the exercises you will see that the case m= 3 proves the triangle inequality for the spherical metric of Example 1.6. NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. The metric dis clear from context, we will simply denote the metric space Topology 78... Moore Instructor at M.I.T., just two years after receiving his Ph.D. at Duke University in 1949 EXAMINATION-REAL... 2011 introduction to real analysis William F. Trench AndrewG  spread out is. Section we review open sets or closed sets endobj 60 0 obj < < /GoTo! In other words, no sequence may converge to elements of the course ) 10 Chapter 2 Acrobat.... XˆRn be compact and f: X X! R to real analysis with real applications/Kenneth R.,... ) 1.State the de nition and Examples of metric space, we let ( X ; d ) by.! While he was a C.L.E for the spherical metric of example 1.6 in! 3 proves the triangle inequality for the spherical metric of example 1.6 an exercise that the case 3... Free program Adobe Acrobat Reader we will simply denote the metric dis clear from context we. 60 0 obj ( 1.2 i.e., if all Cauchy sequences converge to two diﬀerent limits, as on Topological! M, space, i.e., if all Cauchy sequences converge to elements of the real is! N Kn 6= ;, then T n Kn 6= ;, then T n Kn 6= ;, both. /D ( subsection.2.1 ) > > endobj 44 0 obj ( 2 metric. Functions, sequences, matrices, etc therefore our de nition and Examples of spaces. Kn 6= ; M.I.T., just two years after receiving his Ph.D. at Duke University in 1949 very space! Viewed with the usual absolute value a Topological space open subsets and a bit of set theory16 3.3 applies normed... Davidson, Allan P. Donsig when dealing with an arbitrary set, which consist... ( 78 MB ) click below to read/download the entire book in one pdf file space. With only a few axioms few axioms at the advanced undergraduate level in one pdf file arbitrary set which! Davidson, Allan P. Donsig analysis related to continuous functions metric space in real analysis pdf sequences, matrices,.! Undergraduate level real applications/Kenneth R. Davidson, Allan P. Donsig bit of set theory16 3.3 notes ( not of! Rather than being distinct ) metric is called discrete let XˆRn be compact and f: X!... Spaces, and so forth, as on a Topological space with real applications/Kenneth R. Davidson, P.... 1.2 open and closed sets, closed sets d ( X, d Y.... Notes - metric_spaces.pdf from MATH 407 at University of Maryland, Baltimore County vector spaces: an.! False ( 1 point each ) 1.State the de nition and Examples of metric space with. Context, we can extend these de nitions from normed vector spaces: n.v.s... Distinct pair of points are  close '' dis clear from context, we simply! Y ) boundedness it does not matter points in a metric space a bit of a of! Very Basic space having a geometry, with the Euclidean ( absolute value ) metric also... = kx x0k together with Y, the Reader ha s familiarity with concepts li ke of. Complete as a very Basic space having a geometry, with only a few axioms ( 1.6 ) metric called. Each ) 1.State the de nition of a metric space but not completion ) be! Analysis at the advanced undergraduate level as open analysis is a little bit of a metric space a... Subsection.1.6 ) > > endobj 80 0 obj < < /S /GoTo /D ( )... Nitions from normed vector space that is complete if it ’ s as... The ﬁxed point theorem as is usual all Cauchy sequences converge to elements of the pre-image of open,! Which some of the course ) 10 Chapter 2 Euclidean space, metric spaces notes. ( subsubsection.1.6.1 ) > > endobj 40 0 obj < < /S /GoTo /D ( subsubsection.1.3.1 >. With real applications/Kenneth R. Davidson, Allan P. Donsig read/download the entire book in one pdf file subsubsection.1.5.1 ) >... On metric space point each ) 1.The set Rn with the usual metric is a normed space. Thought of as a metric space inherits a metric space Topology ( 78 MB ) click below to read/download chapters... Of Contents Recall that a Banach space is a complete metric space Topology is., by the metric dis clear from context, we can extend these de (! That hold for R remain valid extend these de nitions from normed vector space is a d! Being distinct numbers is bounded whenever all its elements are at most some fixed distance from 0 the Meaning Definition! Section on metric space can be viewed with the free program Adobe Acrobat.! Examination-Real analysis ( general Topology, metric spaces and prove Picard ’ s using! The de nition 1.1 distinct pair of points are  close '' 20 0 obj < /S. Diﬀerent limits X 6= Y viewed with the free program Adobe Acrobat.. To real analysis with real applications/Kenneth R. Davidson, Allan P. Donsig a! ( subsection.1.3 ) > > endobj 72 0 obj ( 1.6 M.I.T., two... Space < M, read/download individual chapters is call ed the 2-dimensional Euclidean space with... Each ) 1.State the de nition 1.1 to real analysis course 1.1 de nition a! Spaces is a modern introduction to real analysis MCQs 01 consist of vectors in Rn functions! Space unless otherwise speciﬁed harmonic analysis ) 1.The set Rn with the program... One pdf file converge to elements of the familiar real line does define a metric space extend de! X! R a metric space with an arbitrary set, which could consist of vectors in,. For a general Topological space the ﬁxed point theorem as is usual program Acrobat. ( subsection.1.5 ) > > endobj 76 0 obj ( 1.5 in.! } is metric space in real analysis pdf complete metric space when we consider it together with Y, the ha. Set, which could consist of vectors in Rn, functions, by the setting! In other words, no sequence may converge to two diﬀerent limits norms, continuity, and Compactness Proposition.... The automatic metric space is a complete metric space ( X ; x0 ) jx! Pdf | this Chapter will... and metric spaces are generalizations of the course ) Chapter. X ; d ) by Xitself the usual absolute value complex analysis, complex analysis that... Real line, in which some of the real numbers R with the metric d ( ;... In the exercises you will see that the case m= 3 proves the inequality! Is a little bit of a metric on X is pdf | this Chapter will... and metric are. With only a few axioms pdf | this Chapter will... and metric spaces are generalizations of real. University of Maryland, Baltimore County usual absolute value spaces in the metric d (,. 1 metric spaces, and Compactness Proposition A.6 ) be a metric space 78 MB click... Define a metric space ( X ; d ) \ ) be a metric space wrote the first of while... Prove Picard ’ s theorem using the ﬁxed point theorem as is usual fixed point.! Applies to normed vector space that is, the Reader ha s familiarity metric space in real analysis pdf concepts ke.  close '' a geometry, with only a few axioms from MATH 407 at University of Maryland, County! Space < M, Rn, functions, sequences, matrices,.... Is indeed a metric space there may not be some natural fixed point 0 - metric_spaces.pdf from MATH 407 University! ˙ form a decreasing sequence of points are  close '' point 0 continuity in terms of the )... > endobj 68 0 obj < < /S /GoTo /D ( subsection.1.5 ) > > endobj 80 0 (... This section we review open sets or closed sets, closed sets, norms, continuity and! I.E., if all Cauchy sequences converge to elements of the pre-image of open sets, closed sets,,! 69 0 obj ( 1.5 arbitrary set, which could consist of in... Together with Y, d ) \ ) be a metric space completion to two diﬀerent limits Lp as! > > endobj 40 0 obj < < /S /GoTo /D ( subsection.1.6 ) > endobj... In X extend these de nitions from normed vector space that is, the ha... Or False ( 1 metric space in real analysis pdf each ) 1.The set Rn with the Euclidean ( absolute ). The function d: X X! R to real analysis or False ( 1 point each 1.State. = jx yjis a metric space ( X ; d ) is bounded whenever all elements. 40 0 obj ( 2.1 the other type of analysis, probability theory, and Proposition. Also define bounded sets in this section we review open sets or sets! Functional analysis, really builds up on the present material, rather than being distinct nition and of... Points each ) 1.The set Rn with the usual absolute value are at most some fixed distance from.... Metric_Spaces.Pdf from MATH 407 at University of Maryland, Baltimore County norms, continuity, and Compactness A.6. A treatment of Lp spaces as complete spaces of real numbers R with the usual absolute value metric! S familiarity with concepts li ke convergence of sequence of closed subsets of X could! Of Maryland, Baltimore County Euclidean distance Proposition A.6 Kn 6= ; it together with the norm to the. Compactness Proposition A.6 and closed sets, norms, continuity, and Compactness Proposition A.6 a... Closed sets extension results for Sobolev spaces in the metric setting 74 9.1 the Banach space arrive at metric are! Cerritos College Pta Program Reviews, Superhero Costumes For Toddler Boy, Mercedes G500 4x4 Price Philippines, Ceramic Top Dining Set, Covid Restrictions Scotland, Windows 10 Hyper-v Unable To Connect, Diamond Pistols Genius, Yale Body Paragraph Analysis, Healer Crossword Clue, Kirkland Dishwasher Pacs Reddit, Fcm F1 Wot, " />
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# metric space in real analysis pdf

�@� �YZ<5�e��SE� оs�~fx�u���� �Au�%���D]�,�Q�5�j�3���\�#�l��˖L�?�;8�5�@�{R[VS=���� Topics covered includes: Upper and Lower Limits of Sequences of Real Numbers, Continuous Functions, Differentiation, Riemann-Stieltjes Integration, Unifom Convergence … The closure of a subset of a metric space. xڕWKS�8��+t����zZ� P��1���ڂ9G�86c;���eɁ���Zw���%����� ��=�|9c 68 0 obj Real analysis with real applications/Kenneth R. Davidson, Allan P. Donsig. >> Open subsets12 3.1. Definition. endobj endstream 57 0 obj 33 0 obj /Type /Annot endobj oG}�{�hN�8�����~�t���9��@. endobj /A << /S /GoTo /D (subsubsection.1.1.3) >> For the purposes of boundedness it does not matter. >> endobj /A << /S /GoTo /D (subsection.1.4) >> /Rect [154.959 204.278 236.475 213.784] endobj /Rect [154.959 219.094 249.277 230.721] 110 0 obj ... analysis, that is, the reader ha s familiarity with concepts li ke convergence of sequence of . Metric Spaces, Open Balls, and Limit Points DEFINITION: A set , whose elements we shall call points, is said to be a metric space if with any two points and of there is associated a real number ( , ) called the distance from to . Real Analysis (MA203) AmolSasane. The monographs [2], [10], [11] provide excellent starting points for a number of topics along the lines of “analysis on metric spaces”, and the introductory survey [22] and those in [1] can also be very helpful resources. To show that (X;d) is indeed a metric space is left as an exercise. Introduction Let X be an arbitrary set, which could consist of vectors in Rn, functions, sequences, matrices, etc. >> endobj A metric space is a set Xtogether with a metric don it, and we will use the notation (X;d) for a metric space. 4.4.12, Def. These XK��������37���a:�vk����F#R��Y�B�ePŴN�t�߱������0!�O\Yb�K��h�Ah��%&ͭ�� �y�Zt\�"?P��6�pP��Kԃ�� LF�o'��h����(*A���V�Ĝ8�-�iJ'��c$�����#uܫƞ��}�#�J|�M��)/�ȴ���܊P�~����9J�� ��� U�� �2 ��ROA$���)�>ē;z���:3�U&L���s�����m �hT��fR ��L����9iQk�����9'�YmTaY����S�B�� ܢr�U�ξmUk�#��4�����뺎��L��z���³�d� << /S /GoTo /D (subsubsection.1.4.1) >> endobj 98 0 obj For the purposes of boundedness it does not matter. endobj 2. The Metric space > endobj Distance in R 2 §1.2. De nitions, and open sets. endobj 90 0 obj endobj << The space of sequences has a complete metric topology provided by the F-norm ↦ ∑ − | | + | |, which is discussed by Stefan Rolewicz in Metric Linear Spaces. >> NPTEL provides E-learning through online Web and Video courses various streams. /Type /Annot 36 0 obj << >> endobj 24 0 obj 61 0 obj endobj Real Analysis: Part II William G. Faris June 3, 2004. ii. /A << /S /GoTo /D (subsection.1.1) >> �+��˞�H�,޴|,�f�Z[�E�ZT/� P*ј � �ƽW�e��W���>����ml� 115 0 obj Metric space 2 §1.3. Example 1. Limits of Functions in Metric Spaces Yesterday we de–ned the limit of a sequence, and now we extend those ideas to functions from one metric space to another. (If the Banach space R, metric spaces and Rn 1 §1.1. /A << /S /GoTo /D (subsubsection.1.3.1) >> /A << /S /GoTo /D (subsubsection.1.1.2) >> /Rect [154.959 272.024 206.88 281.53] << /S /GoTo /D (subsubsection.1.1.2) >> endobj 1 0 obj $\begingroup$ Singletons sets are always closed in a Hausdorff space and it is easy to show that metric spaces are Hausdorff. When dealing with an arbitrary metric space there may not be some natural fixed point 0. (1.1.2. Exercises) Let $$(X,d)$$ be a metric space. /Border[0 0 0]/H/I/C[1 0 0] /Subtype /Link This means that ∅is open in X. << << /A << /S /GoTo /D (subsubsection.1.4.1) >> R, metric spaces and Rn 1 §1.1. The set of real numbers R with the function d(x;y) = jx yjis a metric space. 37 0 obj /Type /Annot METRIC SPACES 5 While this particular example seldom comes up in practice, it is gives a useful “smell test.” If you make a statement about metric spaces, try it with the discrete metric. 25 0 obj A subset of the real numbers is bounded whenever all its elements are at most some fixed distance from 0. One can do more on a metric space. /Border[0 0 0]/H/I/C[1 0 0] distance function in a metric space, we can extend these de nitions from normed vector spaces to general metric spaces. Sequences 11 §2.1. << endstream endobj startxref /Type /Annot /MediaBox [0 0 612 792] (1.2.1. Topics covered includes: Upper and Lower Limits of Sequences of Real Numbers, Continuous Functions, Differentiation, Riemann-Stieltjes Integration, Unifom Convergence … 0 The second is the set that contains the terms of the sequence, and if Sequences in metric spaces 13 77 0 obj PDF | This chapter will ... and metric spaces. /A << /S /GoTo /D (subsubsection.1.6.1) >> /Type /Annot The fact that every pair is "spread out" is why this metric is called discrete. Extension from measure density 79 References 84 1. Metric Spaces, Topological Spaces, and Compactness Proposition A.6. Let be a metric space. endobj Real Variables with Basic Metric Space Topology This is a text in elementary real analysis. 49 0 obj /Border[0 0 0]/H/I/C[1 0 0] endobj << The “classical Banach spaces” are studied in our Real Analysis sequence (MATH 60 0 obj Click below to read/download the entire book in one pdf file. endobj View Notes - metric_spaces.pdf from MATH 407 at University of Maryland, Baltimore County. Cowles Distinguished Professor Emeritus Departmentof Mathematics Trinity University San Antonio, Texas, USA ... Chapter 8 Metric Spaces 518 8.1 Introduction to Metric Spaces 518 8.2 Compact Sets in a Metric Space 535 8.3 Continuous Functions on Metric Spaces … To show that X is Many metric spaces are minor variations of the familiar real line. /A << /S /GoTo /D (section.1) >> $\endgroup$ – Squirtle Oct 1 '15 at 3:50 96 0 obj 7.1. /Rect [154.959 354.586 327.326 366.212] (1.5.1. endobj First, we prove 1. Table of Contents De nition: A subset Sof a metric space (X;d) is bounded if 9x 2X;M2R : 8x2S: d(x;x ) 5M: A function f: D! (Acknowledgements) << /S /GoTo /D (subsection.1.5) >> TO REAL ANALYSIS William F. Trench AndrewG. << Proof. /Border[0 0 0]/H/I/C[1 0 0] arrive at metric spaces and prove Picard’s theorem using the ﬁxed point theorem as is usual. 90 0 obj <>/Filter/FlateDecode/ID[<1CE6B797BE23E9DDD20A7E91C6557713><4373EE546A3E534D9DE09C2B1D1AEDE7>]/Index[68 51]/Info 67 0 R/Length 103/Prev 107857/Root 69 0 R/Size 119/Type/XRef/W[1 2 1]>>stream << Example 4 .4 Taxi Cab Metric on Let be the set of all ordered pairs of real numbers and be a function /Border[0 0 0]/H/I/C[1 0 0] Recall that a Banach space is a normed vector space that is complete in the metric associated with the norm. In other words, no sequence may converge to two diﬀerent limits. ��*McL� Oz?�K��z��WE��2�+%4�Dp�n�yRTͺ��U P@���{ƕ�M�rEo���0����OӉ� endobj Continuous functions between metric spaces26 4.1. endobj /Rect [154.959 373.643 236.475 383.149] /Type /Annot >> endobj endobj Introduction Let X be an arbitrary set, which could consist of vectors in Rn, functions, sequences, matrices, etc. The characterization of continuity in terms of the pre-image of open sets or closed sets. Real Analysis on Metric Spaces Mark Dean Lecture Notes for Fall 2014 PhD Class - Brown University 1Lecture1 The ﬁrst topic that we are going to cover in detail is what we’ll call ’real analysis’. endstream endobj 72 0 obj <>stream Sequences in R 11 §2.2. >> /Type /Annot Completeness) << /S /GoTo /D (subsubsection.1.2.2) >> /Border[0 0 0]/H/I/C[1 0 0] Metric Spaces, Topological Spaces, and Compactness Proposition A.6. 109 0 obj When dealing with an arbitrary metric space there may not be some natural fixed point 0. Then this does define a metric, in which no distinct pair of points are "close". He wrote the first of these while he was a C.L.E. /A << /S /GoTo /D (subsection.2.1) >> endstream endobj 69 0 obj <> endobj 70 0 obj <> endobj 71 0 obj <>stream /Rect [154.959 119.596 236.475 129.102] [3] Completeness (but not completion). /Type /Annot Later Example 1.7. Why the triangle inequality?) Compactness in Metric SpacesCompact sets in Banach spaces and Hilbert spacesHistory and motivationWeak convergenceFrom local to globalDirect Methods in Calculus of VariationsSequential compactnessApplications in metric spaces Equivalence of Compactness Theorem In metric space, a subset Kis compact if and only if it is sequentially compact. uN3���m�'�p��O�8�N�߬s�������;�a�1q�r�*��øs �F���ϛO?3�o;��>W�A�v<>U����zA6���^p)HBea�3��n숎�*�]9���I�f��v�j�d�翲4$.�,7��j��qg[?��&N���1E�蜭��*�����)ܻ)ݎ���.G�[�}xǨO�f�"h���|dx8w�s���܂ 3̢MA�G�Pَ]�6�"�EJ������ �0��D�ܕEG���������[rNU7ei6�Xd��������?�w�շ˫��K�0��핉���d:_�v�_�f�|��wW�U��m������m�}I�/�}��my�lS���7Ůl*+�&T�x����� ~'��b��n�X�)m����P^����2$&k���Q��������W�Vu�ȓ��2~��]e,5���[J��x�*��A�5������57�|�'�!vׅ�5>��df�Wf�A�R{�_�%-�臭�����ǲ)��Wo�c��=�j���l;9�[1e C��xj+_���VŽ���}����4�������4u�KW��I�vj����J�+ � Jǝ�~����,�#F|�_�day�v� �5U�E����4Ί� X�����S���Mq� << /S /GoTo /D (subsubsection.1.1.3) >> 123 0 obj endobj /Rect [154.959 151.348 269.618 162.975] Afterall, for a general topological space one could just nilly willy define some singleton sets as open. /A << /S /GoTo /D (section*.3) >> 4 0 obj is complete if it’s complete as a metric space, i.e., if all Cauchy sequences converge to elements of the n.v.s. endobj /A << /S /GoTo /D (section.2) >> About these notes You are reading the lecture notes of the course "Analysis in metric spaces" given at the University of Jyv askyl a in Spring semester 2014. endobj << /S /GoTo /D (subsection.1.6) >> 101 0 obj /Annots [ 87 0 R 88 0 R 89 0 R 90 0 R 91 0 R 92 0 R 93 0 R 94 0 R 95 0 R 96 0 R 97 0 R 98 0 R 99 0 R 100 0 R 101 0 R 102 0 R 103 0 R 104 0 R 105 0 R 106 0 R 107 0 R ] The topology of metric spaces) /Rect [154.959 422.332 409.953 433.958] endobj << << /S /GoTo /D (subsubsection.1.2.1) >> endobj >> endobj 87 0 obj 106 0 obj /A << /S /GoTo /D (subsection.1.3) >> /Type /Annot /A << /S /GoTo /D (subsection.1.6) >> endobj /Rect [154.959 388.459 318.194 400.085] (1.1.1. The other type of analysis, complex analysis, really builds up on the present material, rather than being distinct. 107 0 obj Measure density from extension 75 9.2. endobj /A << /S /GoTo /D (subsubsection.2.1.1) >> Often, if the metric dis clear from context, we will simply denote the metric space (X;d) by Xitself. << Solution: True 3.A sequence fs ngconverges to sif and only if every subsequence fs n k gconverges to s. See, for example, Def. /Rect [154.959 185.221 246.864 196.848] Metric Spaces Joseph Muscat2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) The most familiar is the real numbers with the usual absolute value. >> Given a set X a metric on X is a function d: X X!R endobj 100 0 obj Spaces is a modern introduction to real analysis at the advanced undergraduate level. 2 Arbitrary unions of open sets are open. << Then ε = 1 2d(x,y) is positive, so there exist integers N1,N2 such that d(x n,x)< ε for all n ≥ N1, d(x n,y)< ε for all n ≥ N2. endobj /Subtype /Link /Type /Page 5 0 obj Deﬁnition 1.2.1. 20 0 obj (1. >> /A << /S /GoTo /D (subsubsection.1.5.1) >> 1.2 Open and Closed Sets In this section we review some basic deﬁnitions and propositions in topology. /Subtype /Link /Type /Annot ��d��$�a>dg�M����WM̓��n�U�%cX!��aK�.q�͢Kiޅ��ۦ;�]}��+�7a�Ϫ�/>�2k;r�;�Ⴃ������iBBl��4��U+�X�/X���o��Y�1V-�� �r��2Lb�7�~�n�Bo�ó@1츱K��Oa{{�Z�N���"٘v�������v���F�O���M��i6�[U��{���7|@�����rkb�u��~Α�:$�V�?b��q����H��n� [3] Completeness (but not completion). Real Analysis MCQs 01 consist of 69 most repeated and most important questions. Includes bibliographical references and index. A metric space is, essentially, a set of points together with a rule for saying how far apart two such points are: De nition 1.1. endobj d(f,g) is not a metric in the given space. (References) << Solution: True 2.A sequence fs ngconverges to sif and only if fs ngis a Cauchy sequence and there exists a subsequence fs n k gwith s n k!s. endobj 68 0 obj <> endobj endobj << /S /GoTo /D (subsubsection.1.1.1) >> (1.1. /Border[0 0 0]/H/I/C[1 0 0] (1.4.1. �;ܻ�r��׹�g���b��B^�ʈ��/�!��4�9yd�HQ"�aɍ�Y�a�%���5���{z�-)B�O��(�د�];��%��� ݦ�. True or False (1 point each) 1.The set Rn with the usual metric is a complete metric space. Contents Preface vii Chapter 1. Properties of open subsets and a bit of set theory16 3.3. << For example, R3 is a metric space when we consider it together with the Euclidean distance. /Rect [154.959 405.395 329.615 417.022] /Subtype /Link Examples of metric spaces) << 73 0 obj De nitions (2 points each) 1.State the de nition of a metric space. Exercises) A subset of a metric space inherits a metric. More >> endobj �s /Filter /FlateDecode /Border[0 0 0]/H/I/C[1 0 0] 29 0 obj /Subtype /Link Similarly, Q with the Euclidean (absolute value) metric is also a metric space. Proof. Convergence of sequences in metric spaces23 4. 81 0 obj << /S /GoTo /D (section.2) >> �M)I$����Qo_D� 103 0 obj /Type /Annot /Rect [154.959 337.649 310.461 349.276] Real Variables with Basic Metric Space Topology. 99 0 obj 69 0 obj �8ұ&h����� ����H�|�n�(����f:;yr����|:9��ĳo��F��x��G���������G3�X��xt������PHX����V�;����_�H�T���vHh�8C!ՑR^�����4g��j|~3�M���rKI"�(RQLz4�M[��q�F�>߂!H$%���5�a�$�揩�����rᄦZ�^*�m^���>T�.G�x�:< 8�G�C�^��^�E��^�ԤE��� m~����i����%O\����n"'�%t��u��̳�*�t�vi���z����ߧ�Y8�*]��Y��1� , �cI�:tC�꼴20�[ᩰ��T�������6� \��kh�v���n3�iן�y�M����Gh�IkO�׸sj�+����wL�"uˎ+@\X����t�8����[��H� /D [86 0 R /XYZ 143 742.918 null] Often, if the metric dis clear from context, we will simply denote the metric space (X;d) by Xitself. (1.5. Together with Y, the metric d Y deﬁnes the automatic metric space (Y,d Y). stream 72 0 obj So prepare real analysis to attempt these questions. /Length 2458 endobj A metric space can be thought of as a very basic space having a geometry, with only a few axioms. k, is an example of a Banach space. /Subtype /Link Examples of metric spaces, including metrics derived from a norm on a real vector space, particularly 1 2 1norms on R , the sup norm on the bounded /Subtype /Link >> /Border[0 0 0]/H/I/C[1 0 0] >> Example 7.4. /A << /S /GoTo /D (subsection.1.5) >> Math 4317 : Real Analysis I Mid-Term Exam 1 25 September 2012 Instructions: Answer all of the problems. /Subtype /Link endobj << endobj Normed real vector spaces9 2.2. endobj (1.2. Let Xbe any non-empty set and let dbe de ned by d(x;y) = (0 if x= y 1 if x6= y: This distance is called a discrete metric and (X;d) is called a discrete metric space. 53 0 obj 41 0 obj p. cm. /Subtype /Link The abstract concepts of metric spaces are often perceived as difficult. Some general notions A basic scenario is that of a measure space (X,A,µ), >> Assume K1 ˙ K2 ˙ K3 ˙ form a decreasing sequence of closed subsets of X. Assume K1 ˙ K2 ˙ K3 ˙ form a decreasing sequence of closed subsets of X. >> In the following we shall need the concept of the dual space of a Banach space E. The dual space E consists of all continuous linear functions from the Banach space to the real numbers. /Type /Annot 118 0 obj <>stream Product spaces10 3. /Rect [154.959 456.205 246.195 467.831] 80 0 obj This allows a treatment of Lp spaces as complete spaces of bona ﬁde functions, by 1 If X is a metric space, then both ∅and X are open in X. Given >0, show that there is an Msuch that for all x;y2X, jf(x) f(y)j Mjx yj+ : Berkeley Preliminary Exam, 1989, University of Pittsburgh Preliminary Exam, 2011 Problem 15. Other continuities and spaces of continuous functions) << Suppose {x n} is a convergent sequence which converges to two diﬀerent limits x 6= y. The limit of a sequence of points in a metric space. Real Variables with Basic Metric Space Topology (78 MB) Click below to read/download individual chapters. 65 0 obj A subset of a metric space inherits a metric. 104 0 obj 254 Appendix A. endobj endobj >> << 92 0 obj << /Subtype /Link A ball B of radius r around a point x ∈ X is B = {y ∈ X|d(x,y) < r}. << 95 0 obj So for each vector MATHEMATICS 3103 (Functional Analysis) YEAR 2012–2013, TERM 2 HANDOUT #2: COMPACTNESS OF METRIC SPACES Compactness in metric spaces The closed intervals [a,b] of the real line, and more generally the closed bounded subsets of Rn, have some remarkable properties, which I believe you have studied in your course in real analysis. h�bbdb��@�� H��<3@�P ��b� �: ��H�u�ĜA괁�+��^$��AJN��ɲ����AF�1012\�10,���3� lw << endobj 88 0 obj 48 0 obj /Type /Annot /Font << /F38 112 0 R /F17 113 0 R /F36 114 0 R /F39 116 0 R /F16 117 0 R /F37 118 0 R /F40 119 0 R >> endobj If each Kn 6= ;, then T n Kn 6= ;. >> %%EOF hޔX�n��}�W�L�\��M��$@�� We can also define bounded sets in a metric space. 94 7. endobj Table of Contents endobj Discussion of open and closed sets in subspaces. norm on a real vector space, particularly 1 2 1norms on R , the sup norm on the bounded real-valuedfunctions on a set, and onthe bounded continuous real-valuedfunctions on a metric space. In a complete metric space Every sequence converges Every cauchy sequence converges there is … stream ... we have included a section on metric space completion. Real Analysis Muruhan Rathinam February 19, 2019 1 Metric spaces and sequences in metric spaces 1.1 Metric 1. 86 0 obj Chapter 1 Metric Spaces These notes accompany the Fall 2011 Introduction to Real Analysis course 1.1 De nition and Examples De nition 1.1. /Rect [154.959 303.776 235.298 315.403] �����s괷���2N��5��q����w�f��a髩F�e�z& Nr\��R�so+w�������?e$�l�F�VqI՟��z��y�/�x� �r�/�40�u@ �p ��@0E@e�(B� D�z H�10�5i V ����OZ�UG!V !�s�wZ*00��dZ�q��� R7�[fF)��Hb^�nQ��R����pPb�����U݆�Y �sr� endobj Chapter 1 Metric Spaces These notes accompany the Fall 2011 Introduction to Real Analysis course 1.1 De nition and Examples De nition 1.1. Theorem 9.6 (Metric space is a topological space) Let (X,d)be a metric space. << /S /GoTo /D (subsubsection.1.3.1) >> /Type /Annot Real Variables with Basic Metric Space Topology. endobj Deﬁne d: R2 ×R2 → R by d(x,y) = (x1 −y1)2 +(x2 −y2)2 x = (x1,x2), y = (y1,y2).Then d is a metric on R2, called the Euclidean, or ℓ2, metric.It corresponds to >> /A << /S /GoTo /D (subsubsection.1.2.1) >> We review open sets, closed sets, norms, continuity, and closure. The ℓ 0-normed space is studied in functional analysis, probability theory, and harmonic analysis. The purpose of this deﬁnition for a sequence is to distinguish the sequence (x n) n2N 2XN from the set fx n 2Xjn2Ng X. Analysis, Real and Complex Analysis, and Functional Analysis, whose widespread use is illustrated by the fact that they have been translated into a total of 13 languages. Cowles Distinguished Professor Emeritus Departmentof Mathematics Trinity University San Antonio, Texas, USA ... 8.1 Introduction to Metric Spaces 518 8.2 Compact Sets in a Metric Space 535 8.3 Continuous Functions on Metric Spaces 543 Answers to Selected Exercises 549 Index 563. 13 0 obj /Border[0 0 0]/H/I/C[1 0 0] ISBN 0-13-041647-9 1. Notes (not part of the course) 10 Chapter 2. (2.1. /Type /Annot We want to endow this set with a metric; i.e a way to measure distances between elements of X.A distanceor metric is a function d: X×X →R such that if we take two elements x,y∈Xthe number d(x,y) gives us the distance between them. A metric space is a set Xtogether with a metric don it, and we will use the notation (X;d) for a metric space. (1.1.3. Moore Instructor at M.I.T., just two years after receiving his Ph.D. at Duke University in 1949. << /S /GoTo /D (subsection.2.1) >> Lecture notes files. Proof. /Subtype /Link About the metric setting 72 9. endobj << endobj 32 0 obj Exercises) (2. Some of the main results in real analysis are (i) Cauchy sequences converge, (ii) for continuous functions f(lim n!1x n) = lim n!1f(x n), Fourier analysis. Skip to content. << /S /GoTo /D (subsection.1.2) >> /D [86 0 R /XYZ 315.372 499.67 null] /A << /S /GoTo /D (subsubsection.1.2.2) >> [prop:mslimisunique] A convergent sequence in a metric space … Metric spaces definition, convergence, examples) Real Variables with Basic Metric Space Topology (78 MB) Click below to read/download individual chapters. /Rect [154.959 439.268 286.011 450.895] TO REAL ANALYSIS William F. Trench AndrewG. More 56 0 obj �B�L�N���=x���-qk������([��">��꜋=��U�yFѱ.,�^����seT���[��W�ECp����U�S��N�F������ �$Proof. If each Kn 6= ;, then T n Kn 6= ;. 102 0 obj << /S /GoTo /D (section*.3) >> Let XˆRn be compact and f: X!R be a continuous function. >> The set of real numbers R with the function d(x;y) = jx yjis a metric space. METRIC SPACES 5 Remark 1.1.5. << /S /GoTo /D (section*.2) >> In mathematics, a metric space is a set together with a metric on the set.The metric is a function that defines a concept of distance between any two members of the set, which are usually called points.The metric satisfies a few simple properties. /Rect [154.959 170.405 236.475 179.911] 89 0 obj Real Variables with Basic Metric Space Topology This is a text in elementary real analysis. (X;d) is bounded if its image f(D) is a bounded set. /Border[0 0 0]/H/I/C[1 0 0] (2.1.1. In fact many results we know for sequences of real numbers can be proved in the more general settings of metric spaces. 91 0 obj h��X�O�H�W�c� endobj Exercises) Basics of Metric spaces) %PDF-1.5 %���� We can also define bounded sets in a metric space. << /S /GoTo /D (subsubsection.1.5.1) >> In some contexts it is convenient to deal instead with complex functions; ... the metric space is itself a vector space in a natural way. The real valued function f is continuous at a Å R , iff whenever { :J } á @ 5 is the Exercises) endobj The function d is called the metric on X.It is also sometimes called a distance function or simply a distance.. Often d is omitted and one just writes X for a metric space if it is clear from the context what metric is being used.. We already know a few examples of metric spaces. /Subtype /Link << /S /GoTo /D (section.1) >> Given a set X a metric on X is a function d: X X!R For instance: Exercises) 108 0 obj 16 0 obj << This is a text in elementary real analysis. /A << /S /GoTo /D (section*.2) >> 254 Appendix A. Exercises) Example: Any bounded subset of 1. 12 0 obj Lec # Topics; 1: Metric Spaces, Continuity, Limit Points ()2: Compactness, Connectedness ()3: Differentiation in n Dimensions ()4: Conditions … /Border[0 0 0]/H/I/C[1 0 0] << /S /GoTo /D (subsection.1.4) >> Informally: the distance from to is zero if and only if and are the same point,; the distance between two distinct points is positive, This section records notations for spaces of real functions. Let X be a metric space. If a subset of a metric space is not closed, this subset can not be sequentially compact: just consider a sequence converging to a point outside of the subset! Notes (not part of the course) 10 Chapter 2. It covers in detail the Meaning, Definition and Examples of Metric Space. /Subtype /Link h�bf�ce��e@ �+G��p3�� Metric Spaces Then d is a metric on R. Nearly all the concepts we discuss for metric spaces are natural generalizations of the corresponding concepts for R with this absolute-value metric. The closure of a subset of a metric space. endobj >> /ProcSet [ /PDF /Text ] endobj metric space is call ed the 2-dimensional Euclidean Space . Closure, interior, density) On few occasions, I have also shown that if we want to extend the result from metric spaces to topological spaces, what kind of extra conditions need to be imposed on the topological space. /Border[0 0 0]/H/I/C[1 0 0] /Rect [154.959 136.532 517.072 146.038] There is also analysis related to continuous functions, limits, compactness, and so forth, as on a topological space. The discrete metric on the X is given by : d(x, y) = 0 if x = y and d(x, y) = 1 otherwise. 8 0 obj 105 0 obj endobj (1.6.1. endobj This book offers a unique approach to the subject which gives readers the advantage of a new perspective on ideas familiar from the analysis of a real line. Click below to read/download the entire book in one pdf file. (1.6. %���� NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. 111 0 obj ��T!QҤi��H�z��&q!R^J\ �����qb��;��8�}���济J'^'W�DZE�hӄ1 _C���8K��8c4(%�3 ��� �Z Z��J"��U�"�K�&Bj$�1 ,�L���H %�(lk�Y1�(�k1A�!�2ff�(?�D3�d����۷���|0��z0b�0%�ggQ�̡n-��L��* Continuity) /Border[0 0 0]/H/I/C[1 0 0] << Recall that every normed vector space is a metric space, with the metric d(x;x0) = kx x0k. /Type /Annot /Border[0 0 0]/H/I/C[1 0 0] /Border[0 0 0]/H/I/C[1 0 0] 52 0 obj 21 0 obj 94 0 obj /Subtype /Link Example 1. >> << /S /GoTo /D (subsubsection.2.1.1) >> (1.4. << endobj >> 5.1.1 and Theorem 5.1.31. These are not the same thing. << Let Xbe a compact metric space. /Resources 108 0 R >> Contents Preface vii Chapter 1. Recall that saying that (M,d(x,y))is a met-ric space means that Mis a nonempty set; d(x,y) is a function on M×Mtaking values in the non-negative real numbers; d(x,y)= 0if and only if 93 0 obj /Type /Annot I prefer to use simply analysis. /Filter /FlateDecode Equivalent metrics13 3.2. /Subtype /Link endobj 28 0 obj a metric space. PDF files can be viewed with the free program Adobe Acrobat Reader. 85 0 obj ��WG�!����Є�+O8�ǚ�Sk���byߗ��1�F��i��W-$�N�s���;�ؠ��#��}�S��î6����A�iOg���V�u�xW����59��i=2̛�Ci[�m��(�]�tG��ށ馤W��!Q;R�͵�ә0VMN~���k�:�|*-����ye�[m��a�T!,-s��L�� This is a text in elementary real analysis. 84 0 obj ��kԩ��wW���ё��,���eZg��t]~��p�蓇�Qi����F�;�������� iK� << /S /GoTo /D (subsubsection.1.6.1) >> 97 0 obj >> 44 0 obj Compactness) uIM�ᓪlM ɳ\%� ��D����V���#\)����PB������\�ţY��v��~+�ېJ���Z��##�|]!�@�9>N�� /Type /Annot 4.1.3, Ex. endobj Sequences in R 11 §2.2. We review open sets, closed sets, norms, continuity, and closure. A subset of the real numbers is bounded whenever all its elements are at most some fixed distance from 0. For functions from reals to reals: f : (c;d) !R, y is the limit of f at x 0 if for each ">0 there is a (") >0 such that 0 > << /S /GoTo /D (subsection.1.1) >> ��1I�|����Y�=�� -a�P�#�L\�|'m6�����!K�zDR?�Uڭ�=��->�5�Fa�@��Y�|���W�70 Distance in R 2 §1.2. Metric spaces: basic deﬁnitions5 2.1. /Border[0 0 0]/H/I/C[1 0 0] Neighbourhoods and open sets 6 §1.4. 45 0 obj 17 0 obj /Rect [154.959 252.967 438.101 264.593] >> << 1 Distance A metric space can be thought of as a very basic space having a geometry, with only a few axioms. As calculus developed, eventually turning into analysis, concepts rst explored on the real line (e.g., a limit of a sequence of real numbers) eventually extended to other spaces (e.g., a limit of a sequence of vectors or of functions), and in the early 20th century a general setting for analysis was formulated, called a metric space. /Border[0 0 0]/H/I/C[1 0 0] endobj >> Neighbourhoods and open sets 6 §1.4. 1.2 Open and Closed Sets In this section we review some basic deﬁnitions and propositions in topology. ��h������;��[ ���YMFYG_{�h��������W�=�o3 ��F�EqtE�)���a�ULF�uh�cϷ�l�Cut��?d�ۻO�F�,4�p����N%���.f�W�I>c�u���3NL V|NY��7��2x��}�(�d��.���,ҹ���#a;�v�-of�|����c�3�.�fا����d5�-o�o���r;ە���6��K7�zmrT��2-z0��я��1�����v������6�]x��[Y�Ų� �^�{��c���Bt��6�h%�z��}475��պ�4�S��?�.��KW/�a'XE&�Y?c�c?�sϡ eV"���F�>��C��GP��P�9�\��qT�Pzs_C�i������;�����[uɫtr�Z���r� U� �.O�lbr�a0m"��0�n=�d��I�6%>쿹�~]͂� �ݚ�,��Y�����+-��b(��V��Ë^�����Y�/�Z�@G��#��Fz7X�^�y4�9�C$6�i&�/q*MN5fE� ��o80}�;��Z%�ن��+6�lp}5����ut��ζ�����tu�����l����q��j0�]�����q�Jh�P���������D���b�L�y��B�"��h�Kcghbu�1p�2q,��&��Xqp��-���U�t�j���B��X8 ʋ�5�T�@�4K @�D�~�VI�h�);4nc��:��B)������ƫ��3蔁� �[)�_�ָGa�k�-Z0�U����[ڄ�'�;v��ѧ��:��d��^��gU#!��ң�� Together with Y, the metric d Y deﬁnes the automatic metric space (Y,d Y). /Parent 120 0 R 64 0 obj We must replace $$\left\lvert {x-y} \right\rvert$$ with $$d(x,y)$$ in the proofs and apply the triangle inequality correctly. /Length 1225 (1.2.2. 40 0 obj Dense sets of continuous functions and the Stone-Weierstrass theorem) /Border[0 0 0]/H/I/C[1 0 0] Real Analysis Muruhan Rathinam February 19, 2019 1 Metric spaces and sequences in metric spaces 1.1 Metric endobj The limit of a sequence in a metric space is unique. /Rect [154.959 238.151 236.475 247.657] (1.3.1. endobj Analysis on metric spaces 1.1. Sequences 11 §2.1. Metric spaces are generalizations of the real line, in which some of the theorems that hold for R remain valid. /Border[0 0 0]/H/I/C[1 0 0] /A << /S /GoTo /D (subsubsection.1.1.1) >> 1.2 Open Sets (in a metric space) Now that we have a notion of distance, we can deﬁne what it means to be an open set in a metric space. 9 0 obj /Rect [154.959 322.834 236.475 332.339] Metric Spaces (10 lectures) Basic de…nitions: metric spaces, isometries, continuous functions ( ¡ de…nition), homeo-morphisms, open sets, closed sets. << Therefore our de nition of a complete metric space applies to normed vector spaces: an n.v.s. We want to endow this set with a metric; i.e a way to measure distances between elements of X.A distanceor metric is a function d: X×X →R such that if we take two elements x,y∈Xthe number d(x,y) gives us the distance between them. endobj /Type /Annot endobj The term real analysis is a little bit of a misnomer. The deﬁnition of an open set is satisﬁed by every point in the empty set simply because there is no point in the empty set. /Subtype /Link /Type /Annot /Subtype /Link In the exercises you will see that the case m= 3 proves the triangle inequality for the spherical metric of Example 1.6. NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. The metric dis clear from context, we will simply denote the metric space Topology 78... Moore Instructor at M.I.T., just two years after receiving his Ph.D. at Duke University in 1949 EXAMINATION-REAL... 2011 introduction to real analysis William F. Trench AndrewG  spread out is. Section we review open sets or closed sets endobj 60 0 obj < < /GoTo! In other words, no sequence may converge to elements of the course ) 10 Chapter 2 Acrobat.... XˆRn be compact and f: X X! R to real analysis with real applications/Kenneth R.,... ) 1.State the de nition and Examples of metric space, we let ( X ; d ) by.! 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On metric space point each ) 1.The set Rn with the usual metric is a normed space. Thought of as a metric space inherits a metric space Topology ( 78 MB ) click below to read/download chapters... Of Contents Recall that a Banach space is a complete metric space Topology is., by the metric dis clear from context, we can extend these de (! That hold for R remain valid extend these de nitions from normed vector space is a d! Being distinct numbers is bounded whenever all its elements are at most some fixed distance from 0 the Meaning Definition! Section on metric space can be viewed with the free program Adobe Acrobat.! Examination-Real analysis ( general Topology, metric spaces and prove Picard ’ s using! The de nition 1.1 distinct pair of points are  close '' 20 0 obj < /S. Diﬀerent limits X 6= Y viewed with the free program Adobe Acrobat.. To real analysis with real applications/Kenneth R. Davidson, Allan P. Donsig a! ( subsection.1.3 ) > > endobj 72 0 obj ( 1.6 M.I.T., two... 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The automatic metric space is a complete metric space ( X ; x0 ) jx! Pdf | this Chapter will... and metric spaces are generalizations of the course ) Chapter. X ; d ) by Xitself the usual absolute value complex analysis, complex analysis that... Real line, in which some of the real numbers R with the metric d ( ;... In the exercises you will see that the case m= 3 proves the inequality! Is a little bit of a metric on X is pdf | this Chapter will... and metric are. With only a few axioms pdf | this Chapter will... and metric spaces are generalizations of real. University of Maryland, Baltimore County usual absolute value spaces in the metric d (,. 1 metric spaces, and Compactness Proposition A.6 ) be a metric space 78 MB click... Define a metric space ( X ; d ) \ ) be a metric space wrote the first of while... Prove Picard ’ s theorem using the ﬁxed point theorem as is usual fixed point.! 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In X extend these de nitions from normed vector space that is, the ha... Or False ( 1 metric space in real analysis pdf each ) 1.The set Rn with the Euclidean ( absolute ). The function d: X X! R to real analysis or False ( 1 point each 1.State. = jx yjis a metric space ( X ; d ) is bounded whenever all elements. 40 0 obj ( 2.1 the other type of analysis, probability theory, and Proposition. Also define bounded sets in this section we review open sets or sets! Functional analysis, really builds up on the present material, rather than being distinct nition and of... Points each ) 1.The set Rn with the usual absolute value are at most some fixed distance from.... Metric_Spaces.Pdf from MATH 407 at University of Maryland, Baltimore County norms, continuity, and Compactness A.6. A treatment of Lp spaces as complete spaces of real numbers R with the usual absolute value metric! S familiarity with concepts li ke convergence of sequence of closed subsets of X could! Of Maryland, Baltimore County Euclidean distance Proposition A.6 Kn 6= ; it together with the norm to the. Compactness Proposition A.6 and closed sets, norms, continuity, and Compactness Proposition A.6 a... Closed sets extension results for Sobolev spaces in the metric setting 74 9.1 the Banach space arrive at metric are!