/Parent 120 0 R /Border[0 0 0]/H/I/C[1 0 0] 100 0 obj 9 0 obj Compactness) endobj /A << /S /GoTo /D (subsubsection.1.1.2) >> Given a set X a metric on X is a function d: X X!R %PDF-1.5
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�B�`L�N���=x���-qk������([��">��꜋=��U�yFѱ.,�^�`���seT���[��W�ECp����U�S��N�F������ �$ 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. /Border[0 0 0]/H/I/C[1 0 0] 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. >> << /S /GoTo /D (subsubsection.1.1.2) >> /A << /S /GoTo /D (subsubsection.1.2.1) >> ��1I�|����Y�=�� -a�P�#�L\�|'m6�����!K�zDR?�Uڭ�=��->�5�Fa�@��Y�|���W�70 Later >> endobj Spaces is a modern introduction to real analysis at the advanced undergraduate level. Real Variables with Basic Metric Space Topology. ... analysis, that is, the reader ha s familiarity with concepts li ke convergence of sequence of . /Subtype /Link Sequences in R 11 §2.2. Distance in R 2 §1.2. A metric space is a set Xtogether with a metric don it, and we will use the notation (X;d) for a metric space. Afterall, for a general topological space one could just nilly willy define some singleton sets as open. endobj PDF files can be viewed with the free program Adobe Acrobat Reader. 1 If X is a metric space, then both ∅and X are open in X. Metric spaces are generalizations of the real line, in which some of the theorems that hold for R remain valid. (1.3.1. (1.1.3. endobj 101 0 obj A metric space consists of a set Xtogether with a function d: X X!R such that: (1) For … /Rect [154.959 238.151 236.475 247.657] 8 0 obj �;ܻ�r���g���b`��B^�ʈ��/�!��4�9yd�HQ"�aɍ�Y�a�%���5�`��{z�-)B�O��(�د�];��%���
ݦ�. Table of Contents METRIC SPACES 5 Remark 1.1.5. 76 0 obj Metric Spaces (10 lectures) Basic de…nitions: metric spaces, isometries, continuous functions ( ¡ de…nition), homeo-morphisms, open sets, closed sets. /A << /S /GoTo /D (section*.3) >> Sequences 11 §2.1. >> �@� �YZ<5�e��SE� оs�~fx�u���� �Au�%���D]�,�Q�5�j�3���\�#�l��˖L�?�;8�5�@�{R[VS=���� Other continuities and spaces of continuous functions) >> /Subtype /Link /A << /S /GoTo /D (subsection.2.1) >> endobj 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 /Border[0 0 0]/H/I/C[1 0 0] NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. A metric space can be thought of as a very basic space having a geometry, with only a few axioms. A subset of a metric space inherits a metric. /Rect [154.959 119.596 236.475 129.102] (1.1.1. endobj >> Let \((X,d)\) be a metric space. endobj 1 0 obj (If the Banach space 80 0 obj 254 Appendix A. /Subtype /Link endobj About the metric setting 72 9. (1.3. Exercises) 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. Example 1. 56 0 obj >> For example, R3 is a metric space when we consider it together with the Euclidean distance. 25 0 obj Real Analysis Muruhan Rathinam February 19, 2019 1 Metric spaces and sequences in metric spaces 1.1 Metric /Subtype /Link Assume K1 ˙ K2 ˙ K3 ˙ form a decreasing sequence of closed subsets of X. endstream endobj d(f,g) is not a metric in the given space. More 0�M�������ϊM���D��"����́_~.pX8�^8�ZGxd0����?�������;ݦ��?�K-H�E��73�s��#b��Wkv�5]��*d����m?ll{i�O!��(�c�.Aԧ�*l�Y$��8�ʗ�O1B�-K�����b�&����r���e�g�0�wV�X/��'2_������|v��٥uM�^��@v���1�m1��^Ύ/�U����c'e-���u�᭠��J�FD�Gl�R���_�0�/
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Click below to read/download the entire book in one pdf file. /Subtype /Link For the purposes of boundedness it does not matter. << /S /GoTo /D (section.2) >> 65 0 obj << /A << /S /GoTo /D (subsection.1.6) >> /Type /Annot 73 0 obj endobj (1.5. 68 0 obj
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Lec # Topics; 1: Metric Spaces, Continuity, Limit Points ()2: Compactness, Connectedness ()3: Differentiation in n Dimensions ()4: Conditions … 45 0 obj The limit of a sequence in a metric space is unique. 52 0 obj PDF | This chapter will ... and metric spaces. endobj Skip to content. /Type /Annot endobj 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. /Type /Annot endobj 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! Real Variables with Basic Metric Space Topology (78 MB) Click below to read/download individual chapters. The most familiar is the real numbers with the usual absolute value. Exercises) 118 0 obj
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(References) 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. << /Border[0 0 0]/H/I/C[1 0 0] Sequences 11 §2.1. 68 0 obj /A << /S /GoTo /D (subsubsection.1.3.1) >> 85 0 obj endobj View Notes - metric_spaces.pdf from MATH 407 at University of Maryland, Baltimore County. /Border[0 0 0]/H/I/C[1 0 0] endobj >> Contents Preface vii Chapter 1. De nitions (2 points each) 1.State the de nition of a metric space. 105 0 obj Exercises) /Border[0 0 0]/H/I/C[1 0 0] << /Subtype /Link >> /Type /Annot /Type /Annot /Rect [154.959 456.205 246.195 467.831] /Border[0 0 0]/H/I/C[1 0 0] endstream
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Open subsets12 3.1. The set of real numbers R with the function d(x;y) = jx yjis a metric space. /Border[0 0 0]/H/I/C[1 0 0] Exercises) A subset of the real numbers is bounded whenever all its elements are at most some fixed distance from 0. >> << Informally: the distance from to is zero if and only if and are the same point,; the distance between two distinct points is positive, hޔX�n��}�W�L�\��M��$@�� (1.2.1. << 98 0 obj 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. Extension results for Sobolev spaces in the metric setting 74 9.1. endobj 40 0 obj 93 0 obj View Notes - metric_spaces.pdf from MATH 407 at University of Maryland, Baltimore County. << /S /GoTo /D (subsection.1.3) >> endobj << /S /GoTo /D (section*.3) >> 106 0 obj A subset of the real numbers is bounded whenever all its elements are at most some fixed distance from 0. When dealing with an arbitrary metric space there may not be some natural fixed point 0. Real Analysis (MA203) AmolSasane. << << /S /GoTo /D (subsubsection.1.1.1) >> 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$.�,7��j��qg[?��&N���1E�蜭��*�����)ܻ)ݎ���.G�[�}xǨO�f�"h���|dx8w�s���܂ 3̢MA�G�Pَ]�6�"�EJ������ The purpose of this deﬁnition for a sequence is to distinguish the sequence (x n) n2N 2XN from the set fx n 2Xjn2Ng X. 21 0 obj 90 0 obj 81 0 obj /Annots [ 87 0 R 88 0 R 89 0 R 90 0 R 91 0 R 92 0 R 93 0 R 94 0 R 95 0 R 96 0 R 97 0 R 98 0 R 99 0 R 100 0 R 101 0 R 102 0 R 103 0 R 104 0 R 105 0 R 106 0 R 107 0 R ] The set of real numbers R with the function d(x;y) = jx yjis a metric space. << /S /GoTo /D (section.1) >> << Recall that a Banach space is a normed vector space that is complete in the metric associated with the norm. For the purposes of boundedness it does not matter. 44 0 obj /A << /S /GoTo /D (subsubsection.1.6.1) >> endobj For functions from reals to reals: f : (c;d) !R, y is the limit of f at x 0 if for each ">0 there is a (") >0 such that 0 > The term real analysis is a little bit of a misnomer. /Type /Annot /Rect [154.959 204.278 236.475 213.784] %���� The discrete metric on the X is given by : d(x, y) = 0 if x = y and d(x, y) = 1 otherwise. To show that X is This section records notations for spaces of real functions. Many metric spaces are minor variations of the familiar real line. A ball B of radius r around a point x ∈ X is B = {y ∈ X|d(x,y) < r}. The Metric space > 254 Appendix A. 17 0 obj Proof. [3] Completeness (but not completion). endobj We review open sets, closed sets, norms, continuity, and closure. Real analysis with real applications/Kenneth R. Davidson, Allan P. Donsig. Introduction Let X be an arbitrary set, which could consist of vectors in Rn, functions, sequences, matrices, etc. �x�mV�aL a�дn�m�ݒ;���Ƞ����b�M���%�
���Pm������Zw���ĵ� �Prif��{6}�0�k��� %�nE�7��,�'&p���)�C��a?�?������{P�Y�8J>��- �O�Ny�D3sq$����TC�b�cW�q�aM Metric space 2 §1.3. /Subtype /Link /Subtype /Link endobj Metric space 2 §1.3. 5 0 obj 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. is complete if it’s complete as a metric space, i.e., if all Cauchy sequences converge to elements of the n.v.s. << /Type /Annot << The limit of a sequence of points in a metric space. << /Subtype /Link To show that (X;d) is indeed a metric space is left as an exercise. /Type /Annot /Subtype /Link Therefore our de nition of a complete metric space applies to normed vector spaces: an n.v.s. Discussion of open and closed sets in subspaces. 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. endobj ��h������;��[
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�[)�_�ָGa�k�-Z0�U����[ڄ�'�;v��ѧ��:��d��^��gU#!��ң�� These Real Variables with Basic Metric Space Topology (78 MB) Click below to read/download individual chapters. >> /Border[0 0 0]/H/I/C[1 0 0] /Filter /FlateDecode (1.2. The closure of a subset of a metric space. endobj So prepare real analysis to attempt these questions. /D [86 0 R /XYZ 143 742.918 null] Extension from measure density 79 References 84 1. h�bbd``b`��@�� H��<3@�P ��b� �: ��H�u�ĜA괁�+��^$��AJN��ɲ����AF�1012\�10,���3� lw
60 0 obj Metric spaces definition, convergence, examples) Let \((X,d)\) be a metric space. << /S /GoTo /D (subsection.1.6) >> 4 0 obj 12 0 obj stream << /S /GoTo /D (subsubsection.1.2.1) >> Normed real vector spaces9 2.2. 107 0 obj /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 >> 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. distance function in a metric space, we can extend these de nitions from normed vector spaces to general metric spaces. /Border[0 0 0]/H/I/C[1 0 0] NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. Exercises) /Resources 108 0 R The real valued function f is continuous at a Å R , iff whenever { :J } á @ 5 is the We can also define bounded sets in a metric space. >> Basics of Metric spaces) Some general notions A basic scenario is that of a measure space (X,A,µ), << /A << /S /GoTo /D (subsubsection.1.1.3) >> h��X�O�H�W�c� >> 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. In a complete metric space Every sequence converges Every cauchy sequence converges there is … >> Example 7.4. Moore Instructor at M.I.T., just two years after receiving his Ph.D. at Duke University in 1949. 96 0 obj Topics covered includes: Upper and Lower Limits of Sequences of Real Numbers, Continuous Functions, Differentiation, Riemann-Stieltjes Integration, Unifom Convergence … Equivalent metrics13 3.2. 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! ISBN 0-13-041647-9 1. /Rect [154.959 151.348 269.618 162.975] 84 0 obj << >> Includes bibliographical references and index. >> /Rect [154.959 288.961 236.475 298.466] /Rect [154.959 252.967 438.101 264.593] << /S /GoTo /D (subsubsection.1.5.1) >> /Rect [154.959 354.586 327.326 366.212] /Type /Annot A subset is called -net if A metric space is called totally bounded if finite -net. Real Analysis MCQs 01 consist of 69 most repeated and most important questions. True or False (1 point each) 1.The set Rn with the usual metric is a complete metric space. /A << /S /GoTo /D (subsection.1.4) >> We can also define bounded sets in a metric space. 99 0 obj /A << /S /GoTo /D (subsection.1.3) >> >> endobj 94 0 obj >> R, metric spaces and Rn 1 §1.1. /Rect [154.959 388.459 318.194 400.085] endobj �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� 72 0 obj /Type /Annot 109 0 obj endobj Similarly, Q with the Euclidean (absolute value) metric is also a metric space. endobj >> endobj /Type /Annot endobj Throughout this section, we let (X,d) be a metric space unless otherwise speciﬁed. /Subtype /Link R, metric spaces and Rn 1 §1.1. 24 0 obj (2. Recall that every normed vector space is a metric space, with the metric d(x;x0) = kx x0k. 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� >> /Rect [154.959 405.395 329.615 417.022] ��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� endobj << /S /GoTo /D (subsubsection.1.3.1) >> (1.6. 88 0 obj /A << /S /GoTo /D (section.2) >> So for each vector << /S /GoTo /D (subsubsection.1.4.1) >> Throughout this section, we let (X,d) be a metric space unless otherwise speciﬁed. /Rect [154.959 272.024 206.88 281.53] >> 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. << /S /GoTo /D [86 0 R /Fit] >> /A << /S /GoTo /D (subsubsection.2.1.1) >> endobj De nitions, and open sets. endobj << /S /GoTo /D (subsection.2.1) >> TO REAL ANALYSIS William F. Trench AndrewG. See, for example, Def. /A << /S /GoTo /D (subsubsection.1.4.1) >> More Contents Preface vii Chapter 1. endobj /A << /S /GoTo /D (subsubsection.1.5.1) >> 115 0 obj Closure, interior, density) Spaces of Functions) 13 0 obj $\endgroup$ – Squirtle Oct 1 '15 at 3:50 (1.2.2. << Exercises) [prop:mslimisunique] A convergent sequence in a metric space … << /S /GoTo /D (section*.2) >> p. cm. Product spaces10 3. Sequences in metric spaces 13 In some contexts it is convenient to deal instead with complex functions; ... the metric space is itself a vector space in a natural way. 33 0 obj (2.1.1. 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 102 0 obj Lecture notes files. Math 4317 : Real Analysis I Mid-Term Exam 1 25 September 2012 Instructions: Answer all of the problems. Properties of open subsets and a bit of set theory16 3.3. endstream
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Metric Spaces, Topological Spaces, and Compactness Proposition A.6. /Subtype /Link 32 0 obj 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. Notes (not part of the course) 10 Chapter 2. For instance: 37 0 obj << /S /GoTo /D (subsubsection.1.1.3) >> 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. endobj /Subtype /Link /Subtype /Link /Filter /FlateDecode 16 0 obj (Acknowledgements) /Type /Annot The characterization of continuity in terms of the pre-image of open sets or closed sets. Table of Contents A subset of a metric space inherits a metric. stream 97 0 obj Analysis on metric spaces 1.1. It covers in detail the Meaning, Definition and Examples of Metric Space. Convergence of sequences in metric spaces23 4. Fourier analysis. Given a set X a metric on X is a function d: X X!R endobj (1.6.1. << Example: Any bounded subset of 1. endobj /Subtype /Link >> endobj These are not the same thing. /ProcSet [ /PDF /Text ] /Border[0 0 0]/H/I/C[1 0 0] %PDF-1.5 77 0 obj Solution: True 3.A sequence fs ngconverges to sif and only if every subsequence fs n k gconverges to s. 1.2 Open and Closed Sets In this section we review some basic deﬁnitions and propositions in topology. Measure density from extension 75 9.2. Together with Y, the metric d Y deﬁnes the automatic metric space (Y,d Y). There is also analysis related to continuous functions, limits, compactness, and so forth, as on a topological space. k, is an example of a Banach space. /Subtype /Link 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. Real Analysis: Part II William G. Faris June 3, 2004. ii. /Type /Annot endobj 108 0 obj ��kԩ��wW���ё��,���eZg��t]~��p�蓇�Qi����F�;�������� iK� Then this does define a metric, in which no distinct pair of points are "close". 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. 28 0 obj Theorem 9.6 (Metric space is a topological space) Let (X,d)be a metric space. 57 0 obj $\begingroup$ Singletons sets are always closed in a Hausdorff space and it is easy to show that metric spaces are Hausdorff. << /S /GoTo /D (subsection.1.4) >> 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 One can do more on a metric space. /Contents 109 0 R Real Variables with Basic Metric Space Topology This is a text in elementary real analysis. /Rect [154.959 170.405 236.475 179.911] 123 0 obj WORKSHEET FOR THE PRELIMINARY EXAMINATION-REAL ANALYSIS (GENERAL TOPOLOGY, METRIC SPACES AND CONTINUITY)3 Problem 14. /Border[0 0 0]/H/I/C[1 0 0] 90 0 obj
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The other type of analysis, complex analysis, really builds up on the present material, rather than being distinct. 7.1. 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. /Border[0 0 0]/H/I/C[1 0 0] NPTEL provides E-learning through online Web and Video courses various streams. 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 /Rect [154.959 136.532 517.072 146.038] 2 Arbitrary unions of open sets are open. /A << /S /GoTo /D (subsection.1.5) >> /Rect [154.959 303.776 235.298 315.403] 36 0 obj 94 7. /A << /S /GoTo /D (subsubsection.1.1.1) >> Deﬁnition 1.2.1. /Subtype /Link The ℓ 0-normed space is studied in functional analysis, probability theory, and harmonic analysis. /Subtype /Link endobj << 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`�� 92 0 obj In other words, no sequence may converge to two diﬀerent limits. /Subtype /Link 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. �s 29 0 obj << /S /GoTo /D (subsubsection.1.2.2) >> /Rect [154.959 373.643 236.475 383.149] 49 0 obj Real Analysis Muruhan Rathinam February 19, 2019 1 Metric spaces and sequences in metric spaces 1.1 Metric 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. /Type /Page Introduction Let X be an arbitrary set, which could consist of vectors in Rn, functions, sequences, matrices, etc. /Length 2458 1.2 Open and Closed Sets In this section we review some basic deﬁnitions and propositions in topology. The fact that every pair is "spread out" is why this metric is called discrete. 61 0 obj endobj (1.4. endobj Often, if the metric dis clear from context, we will simply denote the metric space (X;d) by Xitself. When metric dis understood, we often simply refer to Mas the metric space. /Border[0 0 0]/H/I/C[1 0 0] Example 1.7. << /S /GoTo /D (subsection.1.5) >> This means that ∅is open in X. << Proof. >> >> The abstract concepts of metric spaces are often perceived as difficult. /MediaBox [0 0 612 792] endobj endobj /Type /Annot << /S /GoTo /D (subsection.1.2) >> We must replace \(\left\lvert {x-y} \right\rvert\) with \(d(x,y)\) in the proofs and apply the triangle inequality correctly. `` spread out '' is why this metric is called -net metric space in real analysis pdf metric... Course 1.1 de nition and Examples de nition and Examples of metric space, i.e., all. 1.State the de nition and Examples de nition of a complete metric space applies to normed spaces. 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