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PDF files can be viewed with the free program Adobe Acrobat Reader. One can do more on a metric space. Example 1. << So for each vector /Border[0 0 0]/H/I/C[1 0 0] /Type /Annot View Notes - metric_spaces.pdf from MATH 407 at University of Maryland, Baltimore County. << If each Kn 6= ;, then T n Kn 6= ;. 103 0 obj The abstract concepts of metric spaces are often perceived as difficult. /Type /Annot ��1I�|����Y�=�� -a�P�#�L\�|'m6�����!K�zDR?�Uڭ�=��->�5�Fa�@��Y�|���W�70 endobj 5 0 obj Neighbourhoods and open sets 6 §1.4. >> Assume K1 ˙ K2 ˙ K3 ˙ form a decreasing sequence of closed subsets of X. endobj 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. /Rect [154.959 439.268 286.011 450.895] Let X be a metric space. 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. 104 0 obj /Border[0 0 0]/H/I/C[1 0 0] /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 >> The family Cof subsets of (X,d)defined in Definition 9.10 above satisfies the following four properties, and hence (X,C)is a topological space. << /S /GoTo /D (section.1) >> Real Variables with Basic Metric Space Topology (78 MB) Click below to read/download individual chapters. /Type /Annot A metric space can be thought of as a very basic space having a geometry, with only a few axioms. /Subtype /Link 40 0 obj A subset of a metric space inherits a metric. endobj 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$.�,7��j��qg[?��&N���1E�蜭��*�����)ܻ)ݎ���.G�[�}xǨO�f�"h���|dx8w�s���܂ 3̢MA�G�Pَ]�6�"�EJ������ endobj hޔX�n��}�W�L�\��M��$@�� endobj WORKSHEET FOR THE PRELIMINARY EXAMINATION-REAL ANALYSIS (GENERAL TOPOLOGY, METRIC SPACES AND CONTINUITY)3 Problem 14. Let be a metric space. is complete if it’s complete as a metric space, i.e., if all Cauchy sequences converge to elements of the n.v.s. A metric space is a set Xtogether with a metric don it, and we will use the notation (X;d) for a metric space. Introduction Let X be an arbitrary set, which could consist of vectors in Rn, functions, sequences, matrices, etc. To show that X is 64 0 obj << /S /GoTo /D (subsection.1.5) >> Real Analysis MCQs 01 consist of 69 most repeated and most important questions. Often, if the metric dis clear from context, we will simply denote the metric space (X;d) by Xitself. 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. A metric space is a set Xtogether with a metric don it, and we will use the notation (X;d) for a metric space. 45 0 obj 5.1.1 and Theorem 5.1.31. 115 0 obj The “classical Banach spaces” are studied in our Real Analysis sequence (MATH /Subtype /Link /Type /Annot For instance: Let \((X,d)\) be a metric space. This means that ∅is open in X. h�bbd``b`��@�� H��<3@�P ��b� �: ��H�u�ĜA괁�+��^$��AJN��ɲ����AF�1012\�10,���3� lw
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�ƽW�e��W���>����ml� /Rect [154.959 288.961 236.475 298.466] >> 65 0 obj Skip to content. >> Assume K1 ˙ K2 ˙ K3 ˙ form a decreasing sequence of closed subsets of X. 84 0 obj >> 32 0 obj /A << /S /GoTo /D (subsection.1.4) >> 13 0 obj Exercises) endobj a metric space. /Type /Annot /A << /S /GoTo /D (subsubsection.1.1.1) >> endobj The purpose of this definition for a sequence is to distinguish the sequence (x n) n2N 2XN from the set fx n 2Xjn2Ng X. endobj To show that (X;d) is indeed a metric space is left as an exercise. Afterall, for a general topological space one could just nilly willy define some singleton sets as open. /A << /S /GoTo /D (subsection.1.1) >> The set of real numbers R with the function d(x;y) = jx yjis a metric space. The limit of a sequence of points in a metric space. /A << /S /GoTo /D (subsubsection.1.2.2) >> << /S /GoTo /D (subsubsection.1.2.1) >> 101 0 obj >> �M)I$����Qo_D� Theorem 9.6 (Metric space is a topological space) Let (X,d)be a metric space. The discrete metric on the X is given by : d(x, y) = 0 if x = y and d(x, y) = 1 otherwise. /Border[0 0 0]/H/I/C[1 0 0] 20 0 obj 1 Distance A metric space can be thought of as a very basic space having a geometry, with only a few axioms. << The second is the set that contains the terms of the sequence, and if /Rect [154.959 185.221 246.864 196.848] << xڕWKS�8��+t����zZ� P��1���ڂ9G�86c;���eɁ���Zw���%����� ��=�|9c
endobj /Type /Annot 21 0 obj /Rect [154.959 272.024 206.88 281.53] 98 0 obj << (X;d) is bounded if its image f(D) is a bounded set. distance function in a metric space, we can extend these de nitions from normed vector spaces to general metric spaces. $\endgroup$ – Squirtle Oct 1 '15 at 3:50 Metric space 2 §1.3. 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 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. 68 0 obj
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endobj [prop:mslimisunique] A convergent sequence in a metric space … Metric Spaces Joseph Muscat2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) 107 0 obj Open subsets12 3.1. 85 0 obj Real analysis with real applications/Kenneth R. Davidson, Allan P. Donsig. /MediaBox [0 0 612 792] This allows a treatment of Lp spaces as complete spaces of bona fide functions, by 76 0 obj 60 0 obj Introduction Let X be an arbitrary set, which could consist of vectors in Rn, functions, sequences, matrices, etc. Real Analysis (MA203) AmolSasane. 33 0 obj R, metric spaces and Rn 1 §1.1. For the purposes of boundedness it does not matter. Real Variables with Basic Metric Space Topology This is a text in elementary real analysis. 77 0 obj /Border[0 0 0]/H/I/C[1 0 0] endobj endobj Similarly, Q with the Euclidean (absolute value) metric is also a metric space. /A << /S /GoTo /D (subsection.1.3) >> endobj 48 0 obj endstream
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/Type /Annot Real Variables with Basic Metric Space Topology. 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. 61 0 obj << /S /GoTo /D (subsection.1.2) >> d(f,g) is not a metric in the given space. Example 7.4. 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. Real Analysis: Part II William G. Faris June 3, 2004. ii. Exercises) << /Subtype /Link TO REAL ANALYSIS William F. Trench AndrewG. endobj (1.2.2. /A << /S /GoTo /D (section*.3) >> In some contexts it is convenient to deal instead with complex functions; ... the metric space is itself a vector space in a natural way. A subset of the real numbers is bounded whenever all its elements are at most some fixed distance from 0. 105 0 obj (References) << /S /GoTo /D (section*.2) >> /Filter /FlateDecode Contents Preface vii Chapter 1. We review open sets, closed sets, norms, continuity, and closure. endobj << /S /GoTo /D (section*.3) >> /Subtype /Link endobj Example 1.7. Why the triangle inequality?) /Subtype /Link endobj /D [86 0 R /XYZ 144 720 null] >> >> endobj 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. METRIC SPACES 5 Remark 1.1.5. 24 0 obj endobj Given a set X a metric on X is a function d: X X!R /Rect [154.959 337.649 310.461 349.276] 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. Metric spaces definition, convergence, examples) These are not the same thing. 57 0 obj 1.2 Open and Closed Sets In this section we review some basic definitions and propositions in topology. << 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. endstream Often, if the metric dis clear from context, we will simply denote the metric space (X;d) by Xitself. Let Xbe a compact metric space. /Subtype /Link stream For the purposes of boundedness it does not matter. >> Lecture notes files. NPTEL provides E-learning through online Web and Video courses various streams. The topology of metric spaces) Example 4 .4 Taxi Cab Metric on Let be the set of all ordered pairs of real numbers and be a function I prefer to use simply analysis. Sequences in metric spaces 13 /Border[0 0 0]/H/I/C[1 0 0] Metric Spaces (10 lectures) Basic de…nitions: metric spaces, isometries, continuous functions ( ¡ de…nition), homeo-morphisms, open sets, closed sets. ��WG�!����Є�+O8�ǚ�Sk���byߗ��1�F��i��W-$�N�s���;�ؠ��#��}�S��î6����A�iOg���V�u�xW����59��i=2̛�Ci[�m��(�]�tG��ށ馤W��!Q;R�͵�ә0VMN~���k�:�|*-����ye�[m��a�T!,-s��L�� We can also define bounded sets in a metric space. 4.4.12, Def. ri��֍5O�~G�����aP�{���s3^�v/:0Y�y�ۆ�ԏ�̌�1�Uǭw�D ə�t�SNe���}�̅��l��ʅ$[���Ȑ8kd�C��eH�E[\���\��z��S� $O�
[3] Completeness (but not completion). TO REAL ANALYSIS William F. Trench AndrewG. A subset of the real numbers is bounded whenever all its elements are at most some fixed distance from 0. %���� endobj 88 0 obj endobj endobj (1.6.1. The term real analysis is a little bit of a misnomer. 1. Real Variables with Basic Metric Space Topology This is a text in elementary real analysis. Later endobj >> << /S /GoTo /D [86 0 R /Fit] >> Metric Spaces, Topological Spaces, and Compactness Proposition A.6. /Subtype /Link /Resources 108 0 R ��h������;��[
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�[)�_�ָGa�k�-Z0�U����[ڄ�'�;v��ѧ��:��d��^��gU#!��ң�� << /S /GoTo /D (subsection.2.1) >> Together with Y, the metric d Y defines the automatic metric space (Y,d Y). p. cm. /Border[0 0 0]/H/I/C[1 0 0] Cowles Distinguished Professor Emeritus Departmentof Mathematics Trinity University San Antonio, Texas, USA ... Chapter 8 Metric Spaces 518 8.1 Introduction to Metric Spaces 518 8.2 Compact Sets in a Metric Space 535 8.3 Continuous Functions on Metric Spaces … endobj Define d: R2 ×R2 → R by d(x,y) = (x1 −y1)2 +(x2 −y2)2 x = (x1,x2), y = (y1,y2).Then d is a metric on R2, called the Euclidean, or ℓ2, metric.It corresponds to endobj /Rect [154.959 204.278 236.475 213.784] Metric spaces are generalizations of the real line, in which some of the theorems that hold for R remain valid. >> xڕ˒�6��P�e�*�&� kkv�:�MbWœ��䀡 �e���1����(Q����h�F��갊V߽z{����$Z��0�Z��W*IVF�H���n�9��[U�Q|���Oo����4 ެ�"����?��i���^EB��;]�TQ!�t�u���@Q)�H��/M��S�vwr��#���TvM`�� endobj Metric space 2 §1.3. << << /A << /S /GoTo /D (subsubsection.1.1.3) >> The ℓ 0-normed space is studied in functional analysis, probability theory, and harmonic analysis. 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. ��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� �x�mV�aL a�дn�m�ݒ;���Ƞ����b�M���%�
���Pm������Zw���ĵ� �Prif��{6}�0�k��� %�nE�7��,�'&p���)�C��a?�?������{P�Y�8J>��- �O�Ny�D3sq$����TC�b�cW�q�aM Click below to read/download the entire book in one pdf file. Table of Contents Exercises) >> We review open sets, closed sets, norms, continuity, and closure. If a subset of a metric space is not closed, this subset can not be sequentially compact: just consider a sequence converging to a point outside of the subset! endobj endobj Contents Preface vii Chapter 1. (1.4. 86 0 obj A ball B of radius r around a point x ∈ X is B = {y ∈ X|d(x,y) < r}. 69 0 obj endobj Example: Any bounded subset of 1. A metric space is, essentially, a set of points together with a rule for saying how far apart two such points are: De nition 1.1. Completeness) << NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. << (1.2. 56 0 obj 89 0 obj �����s괷���2N��5��q����w�f��a髩F�e�z& Nr\��R�so+w�������?e$�l�F�VqI՟��z��y�/�x� �r�/�40�u@ �p ��@0E@e�(B� D�z H�10�5i V ����OZ�UG!V
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%�(lk�Y1`�(�k1A�!�2ff�(?�D3�d����۷���|0��z0b�0%�ggQ�̡n-��L��* endobj /Type /Annot There is also analysis related to continuous functions, limits, compactness, and so forth, as on a topological space. Neighbourhoods and open sets 6 §1.4. (2.1.1. /Border[0 0 0]/H/I/C[1 0 0] >> Definition. << /S /GoTo /D (subsubsection.1.2.2) >> 254 Appendix A. endobj endobj Distance in R 2 §1.2. (1.1. Suppose {x n} is a convergent sequence which converges to two different limits x 6= y. About the metric setting 72 9. /Subtype /Link (1.5.1. 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. endobj Fourier analysis. The real valued function f is continuous at a Å R , iff whenever { :J } á @ 5 is the ... we have included a section on metric space completion. PDF | This chapter will ... and metric spaces. For example, R3 is a metric space when we consider it together with the Euclidean distance. The fact that every pair is "spread out" is why this metric is called discrete. Convergence of sequences in metric spaces23 4. In fact many results we know for sequences of real numbers can be proved in the more general settings of metric spaces. 92 0 obj << 81 0 obj In other words, no sequence may converge to two different limits. 44 0 obj 111 0 obj << NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. The most familiar is the real numbers with the usual absolute value. (If the Banach space >> Proof. Extension results for Sobolev spaces in the metric setting 74 9.1. >> 1 If X is a metric space, then both ∅and X are open in X. 8 0 obj The limit of a sequence in a metric space is unique. endobj Includes bibliographical references and index. This is a text in elementary real analysis. Example 1. This is a text in elementary real analysis. Recall that every normed vector space is a metric space, with the metric d(x;x0) = kx x0k. A subset is called -net if A metric space is called totally bounded if finite -net. /Rect [154.959 456.205 246.195 467.831] /Border[0 0 0]/H/I/C[1 0 0] << << Discussion of open and closed sets in subspaces. 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. (1.5. PDF files can be viewed with the free program Adobe Acrobat Reader. �0��D�ܕEG���������[rNU7ei6�Xd��������?�`w�շ˫��K�0��핉���d:_�v�_�f�|��wW�U��m������m�}I�/�}��my�lS���7Ůl*+�&T�x�����
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/A << /S /GoTo /D (subsubsection.1.1.2) >> The closure of a subset of a metric space. Compactness) Math 4317 : Real Analysis I Mid-Term Exam 1 25 September 2012 Instructions: Answer all of the problems. endobj /Type /Annot More /Border[0 0 0]/H/I/C[1 0 0] Therefore our de nition of a complete metric space applies to normed vector spaces: an n.v.s. endobj (1.1.3. >> Some general notions A basic scenario is that of a measure space (X,A,µ), ISBN 0-13-041647-9 1. >> Many metric spaces are minor variations of the familiar real line. 53 0 obj << /A << /S /GoTo /D (subsubsection.1.5.1) >> 123 0 obj 17 0 obj arrive at metric spaces and prove Picard’s theorem using the fixed point theorem as is usual. << /S /GoTo /D (subsection.1.3) >> Discussion of open and closed sets in subspaces. /Type /Annot So prepare real analysis to attempt these questions. (1.1.1. These Sequences 11 §2.1. /A << /S /GoTo /D (subsection.2.1) >> 118 0 obj
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endobj 87 0 obj In the exercises you will see that the case m= 3 proves the triangle inequality for the spherical metric of Example 1.6. endobj /A << /S /GoTo /D (subsubsection.1.6.1) >> /Subtype /Link 2. Recall that a Banach space is a normed vector space that is complete in the metric associated with the norm. (1.3. First, we prove 1. Euclidean metric. 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. 0
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, �cI�:tC�꼴20�[ᩰ��T�������6� \��kh�v���n3�iן�y�M����Gh�IkO�sj�+����wL�"uˎ+@\X����t�8����[��H� /Rect [154.959 388.459 318.194 400.085] 2 Arbitrary unions of open sets are open. In a complete metric space Every sequence converges Every cauchy sequence converges there is … endobj endobj Click below to read/download the entire book in one pdf file. >> h�b```f``�c`e`��e`@ �+G��p3�� De nitions (2 points each) 1.State the de nition of a metric space. Together with Y, the metric d Y defines the automatic metric space (Y,d Y). 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! 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. Metric spaces: basic definitions5 2.1. endobj /Subtype /Link /Border[0 0 0]/H/I/C[1 0 0] Exercises) 28 0 obj %PDF-1.5
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endobj << << /S /GoTo /D (subsubsection.2.1.1) >> endobj Throughout this section, we let (X,d) be a metric space unless otherwise specified. 72 0 obj << 94 7. This section records notations for spaces of real functions. Proof. 36 0 obj Informally: the distance from to is zero if and only if and are the same point,; the distance between two distinct points is positive, Measure density from extension 75 9.2. /Rect [154.959 238.151 236.475 247.657] 108 0 obj (2. Closure, interior, density) /Subtype /Link �B�`L�N���=x���-qk������([��">��꜋=��U�yFѱ.,�^�`���seT���[��W�ECp����U�S��N�F������ �$ endobj Metric Spaces, Topological Spaces, and Compactness Proposition A.6. Continuity) endstream
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>> 41 0 obj Topics covered includes: Upper and Lower Limits of Sequences of Real Numbers, Continuous Functions, Differentiation, Riemann-Stieltjes Integration, Unifom Convergence … $\begingroup$ 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. View Notes - metric_spaces.pdf from MATH 407 at University of Maryland, Baltimore County. << Real Variables with Basic Metric Space Topology. 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� (2.1. /Parent 120 0 R 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. /Border[0 0 0]/H/I/C[1 0 0] /Border[0 0 0]/H/I/C[1 0 0] << /Length 1225 Distance in R 2 §1.2. Spaces of Functions) 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 > Notes (not part of the course) 10 Chapter 2. 254 Appendix A. /Border[0 0 0]/H/I/C[1 0 0] 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), >> /Rect [154.959 322.834 236.475 332.339] /Type /Annot Spaces is a modern introduction to real analysis at the advanced undergraduate level. /Rect [154.959 354.586 327.326 366.212] The Metric space > << k, is an example of a Banach space. >> << 12 0 obj ��*McL� Oz?�K��z��WE��2�+%4�Dp�n�yRTͺ��U P@���{ƕ�M�rEo���0����OӉ� We must replace \(\left\lvert {x-y} \right\rvert\) with \(d(x,y)\) in the proofs and apply the triangle inequality correctly. /Subtype /Link /A << /S /GoTo /D (subsection.1.2) >> Real Analysis Muruhan Rathinam February 19, 2019 1 Metric spaces and sequences in metric spaces 1.1 Metric /Subtype /Link 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 . 4 0 obj When dealing with an arbitrary metric space there may not be some natural fixed point 0. 100 0 obj The set of real numbers R with the function d(x;y) = jx yjis a metric space. endobj 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. << /S /GoTo /D (subsubsection.1.1.2) >> 37 0 obj Table of Contents /Subtype /Link /Type /Annot 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’. << 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. /Type /Annot /Subtype /Link A metric space consists of a set Xtogether with a function d: X X!R such that: (1) For … Lec # Topics; 1: Metric Spaces, Continuity, Limit Points ()2: Compactness, Connectedness ()3: Differentiation in n Dimensions ()4: Conditions … /Type /Annot (1.6. 68 0 obj >> /Rect [154.959 373.643 236.475 383.149] He wrote the first of these while he was a C.L.E. Real Analysis (MA203) AmolSasane. 1 Prelude to Modern Analysis 1 1.1 Introduction 1 1.2 Sets and numbers 3 1.3 Functions or mappings 10 1.4 Countability 14 1.5 Point sets 20 1.6 Open and closed sets 28 1.7 Sequences 32 1.8 Series 44 1.9 Functions of a real variable 52 1.10 Uniform convergence 59 1.11 Some linear algebra 69 1.12 Setting off 83 2 Metric Spaces 84 << /S /GoTo /D (subsubsection.1.4.1) >> h��X�O�H�W�c� (1.4.1. 4.1.3, Ex. Moore Instructor at M.I.T., just two years after receiving his Ph.D. at Duke University in 1949. >> /Type /Page 73 0 obj /Type /Annot << /S /GoTo /D (subsection.1.4) >> << /S /GoTo /D (subsubsection.1.5.1) >> Let \((X,d)\) be a metric space. /Rect [154.959 119.596 236.475 129.102] /Border[0 0 0]/H/I/C[1 0 0] 16 0 obj /Border[0 0 0]/H/I/C[1 0 0] endobj /A << /S /GoTo /D (subsection.1.5) >> Proof. /Border[0 0 0]/H/I/C[1 0 0] Exercises) 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. /Rect [154.959 405.395 329.615 417.022] Exercises) [3] Completeness (but not completion). endobj << /S /GoTo /D (subsection.1.6) >> Real Variables with Basic Metric Space Topology (78 MB) Click below to read/download individual chapters. Solution: True 3.A sequence fs ngconverges to sif and only if every subsequence fs n k gconverges to s. endobj endobj 95 0 obj >> stream >> Sequences in R 11 §2.2. << True or False (1 point each) 1.The set Rn with the usual metric is a complete metric space. /A << /S /GoTo /D (section.2) >> (1.3.1. /Rect [154.959 151.348 269.618 162.975] << /S /GoTo /D (subsubsection.1.1.3) >> 99 0 obj endobj ... analysis, that is, the reader ha s familiarity with concepts li ke convergence of sequence of . metric space is call ed the 2-dimensional Euclidean Space . 25 0 obj /Border[0 0 0]/H/I/C[1 0 0] Exercises) >> /Subtype /Link endobj Let Xbe a compact metric space. 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. /A << /S /GoTo /D (section.1) >> It covers in detail the Meaning, Definition and Examples of Metric Space. (1.1.2. It is forward-looking in the sense that it first and foremost aims to provide students with the concepts and techniques they need in order to follow more advanced courses in mathematical analysis and neighboring fields. �;ܻ�r���g���b`��B^�ʈ��/�!��4�9yd�HQ"�aɍ�Y�a�%���5�`��{z�-)B�O��(�د�];��%���
ݦ�. /Subtype /Link 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. Let XˆRn be compact and f: X!R be a continuous function. endobj 93 0 obj << /Border[0 0 0]/H/I/C[1 0 0] Sequences 11 §2.1. If each Kn 6= ;, then T n Kn 6= ;. /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 ] Product spaces10 3. endobj /Type /Annot 1 0 obj /Filter /FlateDecode (1.2.1. endobj (Acknowledgements) endobj More 80 0 obj Examples of metric spaces) Notes (not part of the course) 10 Chapter 2. 91 0 obj The characterization of continuity in terms of the pre-image of open sets or closed sets. (1. << See, for example, Def. We can also define bounded sets in a metric space. endobj 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. 106 0 obj The other type of analysis, complex analysis, really builds up on the present material, rather than being distinct. /Type /Annot Continuous functions between metric spaces26 4.1. Equivalent metrics13 3.2. Definition 1.2.1. 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. The space of sequences has a complete metric topology provided by the F-norm ↦ ∑ − | | + | |, which is discussed by Stefan Rolewicz in Metric Linear Spaces. endobj Other continuities and spaces of continuous functions) /Subtype /Link Sequences in R 11 §2.2. R, metric spaces and Rn 1 §1.1. endobj endobj endstream
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