uIM�ᓪlM ɳ\%� ��D����V���#\)����PB������\�ţY��v��~+�ېJ���Z��##�|]!�@�9>N�� >> Real Analysis (MA203) AmolSasane. If each Kn 6= ;, then T n Kn 6= ;. The second is the set that contains the terms of the sequence, and if /Contents 109 0 R A metric space is a set Xtogether with a metric don it, and we will use the notation (X;d) for a metric space. endstream
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1.2 Open and Closed Sets In this section we review some basic deﬁnitions and propositions in topology. /Rect [154.959 272.024 206.88 281.53] endobj Lec # Topics; 1: Metric Spaces, Continuity, Limit Points ()2: Compactness, Connectedness ()3: Differentiation in n Dimensions ()4: Conditions … endobj ��1I�|����Y�=�� -a�P�#�L\�|'m6�����!K�zDR?�Uڭ�=��->�5�Fa�@��Y�|���W�70 /Rect [154.959 405.395 329.615 417.022] Many metric spaces are minor variations of the familiar real line. endobj /Type /Annot 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`�� 64 0 obj Normed real vector spaces9 2.2. /Rect [154.959 337.649 310.461 349.276] For the purposes of boundedness it does not matter. /A << /S /GoTo /D (subsection.1.4) >> Real Variables with Basic Metric Space Topology. To show that (X;d) is indeed a metric space is left as an exercise. 33 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 ] Exercises) >> Other continuities and spaces of continuous functions) endobj 44 0 obj /Type /Annot /A << /S /GoTo /D (subsubsection.2.1.1) >> Spaces of Functions) These are not the same thing. Continuous functions between metric spaces26 4.1. /Subtype /Link Convergence of sequences in metric spaces23 4. endobj endobj Exercises) endobj /MediaBox [0 0 612 792] 84 0 obj De nitions, and open sets. 52 0 obj 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 … Closure, interior, density) << /S /GoTo /D (subsection.1.1) >> Let Xbe a compact metric space. 87 0 obj endobj A subset of the real numbers is bounded whenever all its elements are at most some fixed distance from 0. Throughout this section, we let (X,d) be a metric space unless otherwise speciﬁed. Exercises) endobj endobj /Type /Annot Real Analysis Muruhan Rathinam February 19, 2019 1 Metric spaces and sequences in metric spaces 1.1 Metric 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! Example 4 .4 Taxi Cab Metric on Let be the set of all ordered pairs of real numbers and be a function 90 0 obj 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! >> The other type of analysis, complex analysis, really builds up on the present material, rather than being distinct. 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 is a text in elementary real analysis. �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�$�揩�����rᄦZ�^*�m^���>T�.G�x�:< 8�G�C�^��^�E��^�ԤE��� m~����i���`�%O\����n"'�%t��u`��̳�*�t�vi���z����ߧ�Y8�*]��Y��1�
, �cI�:tC�꼴20�[ᩰ��T�������6� \��kh�v���n3�iן�y�M����Gh�IkO�sj�+����wL�"uˎ+@\X����t�8����[��H� 4.4.12, Def. (1.1.1. Exercises) /A << /S /GoTo /D (section*.2) >> 12 0 obj Product spaces10 3. endobj Deﬁnition 1.2.1. 85 0 obj endobj >> The characterization of continuity in terms of the pre-image of open sets or closed sets. /Subtype /Link We review open sets, closed sets, norms, continuity, and closure. Dense sets of continuous functions and the Stone-Weierstrass theorem) << Often, if the metric dis clear from context, we will simply denote the metric space (X;d) by Xitself. 254 Appendix A. /Rect [154.959 373.643 236.475 383.149]
86 0 obj Real Variables with Basic Metric Space Topology (78 MB) Click below to read/download individual chapters. /Border[0 0 0]/H/I/C[1 0 0] There is also analysis related to continuous functions, limits, compactness, and so forth, as on a topological space. Topics covered includes: Upper and Lower Limits of Sequences of Real Numbers, Continuous Functions, Differentiation, Riemann-Stieltjes Integration, Unifom Convergence … Some of the main results in real analysis are (i) Cauchy sequences converge, (ii) for continuous functions f(lim n!1x n) = lim n!1f(x n), endobj /Rect [154.959 422.332 409.953 433.958] Then this does define a metric, in which no distinct pair of points are "close". /Subtype /Link ... we have included a section on metric space completion. 8 0 obj ə�t�SNe���}�̅��l��ʅ$[���Ȑ8kd�C��eH�E[\���\��z��S� $O�
Theorem 9.6 (Metric space is a topological space) Let (X,d)be a metric space. /Border[0 0 0]/H/I/C[1 0 0] Metric Spaces Joseph Muscat2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) << /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 >> %PDF-1.5
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/Border[0 0 0]/H/I/C[1 0 0] 57 0 obj Proof. Continuity) A subset of a metric space inherits a metric. /Type /Annot /Rect [154.959 151.348 269.618 162.975] Analysis on metric spaces 1.1. 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. /Border[0 0 0]/H/I/C[1 0 0] endobj endobj endobj endobj /Resources 108 0 R %PDF-1.5 stream 80 0 obj Introduction Let X be an arbitrary set, which could consist of vectors in Rn, functions, sequences, matrices, etc. 1.2 Open and Closed Sets In this section we review some basic deﬁnitions and propositions in topology. In a complete metric space Every sequence converges Every cauchy sequence converges there is … First, we prove 1. << /Type /Annot endobj Afterall, for a general topological space one could just nilly willy define some singleton sets as open. The real valued function f is continuous at a Å R , iff whenever { :J } á @ 5 is the This allows a treatment of Lp spaces as complete spaces of bona ﬁde functions, by Click below to read/download the entire book in one pdf file. /D [86 0 R /XYZ 144 720 null] /Rect [154.959 204.278 236.475 213.784] /Type /Annot Definition. 91 0 obj /A << /S /GoTo /D (subsection.1.1) >> $\endgroup$ – Squirtle Oct 1 '15 at 3:50 endobj Given a set X a metric on X is a function d: X X!R Sequences 11 §2.1. /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’. /Border[0 0 0]/H/I/C[1 0 0] The limit of a sequence in a metric space is unique. endobj 5 0 obj << /S /GoTo /D (subsection.1.5) >> << /S /GoTo /D (subsubsection.1.2.1) >> /Rect [154.959 322.834 236.475 332.339] /A << /S /GoTo /D (subsubsection.1.2.1) >> Often, if the metric dis clear from context, we will simply denote the metric space (X;d) by Xitself. 1. Contents Preface vii Chapter 1. endobj The closure of a subset of a metric space. /Rect [154.959 136.532 517.072 146.038] 69 0 obj >> << 9 0 obj << /S /GoTo /D (subsubsection.1.5.1) >> Metric space 2 §1.3. 95 0 obj << /S /GoTo /D (subsection.2.1) >> (1.4. endobj 100 0 obj %���� The fact that every pair is "spread out" is why this metric is called discrete. The purpose of this deﬁnition for a sequence is to distinguish the sequence (x n) n2N 2XN from the set fx n 2Xjn2Ng X. 254 Appendix A. << /S /GoTo /D (subsubsection.1.1.2) >> endobj /Filter /FlateDecode 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. 56 0 obj (1.1. One can do more on a metric space. 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. 109 0 obj endobj /Border[0 0 0]/H/I/C[1 0 0] << /S /GoTo /D (subsection.1.3) >> << /S /GoTo /D (subsubsection.1.2.2) >> k, is an example of a Banach space. 72 0 obj R, metric spaces and Rn 1 §1.1. 94 0 obj << Why the triangle inequality?) 104 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. �����s괷���2N��5��q����w�f��a髩F�e�z& Nr\��R�so+w�������?e$�l�F�VqI՟��z��y�/�x� �r�/�40�u@ �p ��@0E@e�(B� D�z H�10�5i V ����OZ�UG!V
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The monographs [2], [10], [11] provide excellent starting points for a number of topics along the lines of “analysis on metric spaces”, and the introductory survey [22] and those in [1] can also be very helpful resources. More norm on a real vector space, particularly 1 2 1norms on R , the sup norm on the bounded real-valuedfunctions on a set, and onthe bounded continuous real-valuedfunctions on a metric space. ri��֍5O�~G�����aP�{���s3^�v/:0Y�y�ۆ�ԏ�̌�1�Uǭw�D 1 Distance A metric space can be thought of as a very basic space having a geometry, with only a few axioms. Example: Any bounded subset of 1. 68 0 obj endobj /Rect [154.959 119.596 236.475 129.102] He wrote the first of these while he was a C.L.E. << 32 0 obj /A << /S /GoTo /D (subsubsection.1.1.1) >> /Type /Annot (1.1.3. The discrete metric on the X is given by : d(x, y) = 0 if x = y and d(x, y) = 1 otherwise. endobj Neighbourhoods and open sets 6 §1.4. endobj On few occasions, I have also shown that if we want to extend the result from metric spaces to topological spaces, what kind of extra conditions need to be imposed on the topological space. d(f,g) is not a metric in the given space. 73 0 obj /Subtype /Link True or False (1 point each) 1.The set Rn with the usual metric is a complete metric space. 108 0 obj endobj R, metric spaces and Rn 1 §1.1. (1.2. /Type /Annot Open subsets12 3.1. ��*McL� Oz?�K��z��WE��2�+%4�Dp�n�yRTͺ��U P@���{ƕ�M�rEo���0����OӉ� 77 0 obj /A << /S /GoTo /D (subsubsection.1.6.1) >> << /S /GoTo /D (subsubsection.1.3.1) >> Metric spaces are generalizations of the real line, in which some of the theorems that hold for R remain valid. [3] Completeness (but not completion). Metric Spaces, Topological Spaces, and Compactness Proposition A.6. endobj << uN3���m�'�p��O�8�N�߬s�������;�a�1q�r�*��øs
�F���ϛO?3�o;��>W�A�v<>U����zA6���^p)HBea�3��n숎�*�]9���I�f��v�j�d�翲4$.�,7��j��qg[?��&N���1E�蜭��*�����)ܻ)ݎ���.G�[�}xǨO�f�"h���|dx8w�s���܂ 3̢MA�G�Pَ]�6�"�EJ������ 17 0 obj 98 0 obj 7.1. It covers in detail the Meaning, Definition and Examples of Metric Space. << /S /GoTo /D (subsubsection.1.4.1) >> >> 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. Sequences 11 §2.1. 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. /A << /S /GoTo /D (subsubsection.1.1.3) >> /Rect [154.959 303.776 235.298 315.403] (2. 36 0 obj metric space is call ed the 2-dimensional Euclidean Space . Table of Contents The topology of metric spaces) (1.1.2. >> 16 0 obj /Border[0 0 0]/H/I/C[1 0 0] 20 0 obj Assume K1 ˙ K2 ˙ K3 ˙ form a decreasing sequence of closed subsets of X. Later /Border[0 0 0]/H/I/C[1 0 0] endobj For example, R3 is a metric space when we consider it together with the Euclidean distance. endobj >> endobj Let \((X,d)\) be a metric space. 0
(1.6.1. Some general notions A basic scenario is that of a measure space (X,A,µ), endobj Fourier analysis. >> Lecture notes files. More (1.6. (1.2.1. 4.1.3, Ex. endobj Sequences in R 11 §2.2. /Border[0 0 0]/H/I/C[1 0 0] /Border[0 0 0]/H/I/C[1 0 0] Let Xbe a compact metric space. endobj endobj Moore Instructor at M.I.T., just two years after receiving his Ph.D. at Duke University in 1949. endobj << The “classical Banach spaces” are studied in our Real Analysis sequence (MATH /Rect [154.959 252.967 438.101 264.593] (1. Discussion of open and closed sets in subspaces. The limit of a sequence of points in a metric space. 94 7. Exercises) These 115 0 obj endobj �@� �YZ<5�e��SE� оs�~fx�u���� �Au�%���D]�,�Q�5�j�3���\�#�l��˖L�?�;8�5�@�{R[VS=���� /A << /S /GoTo /D (subsection.1.5) >> endobj endobj PDF files can be viewed with the free program Adobe Acrobat Reader. << 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 >> 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. Spaces is a modern introduction to real analysis at the advanced undergraduate level. 102 0 obj (References) Real Variables with Basic Metric Space Topology (78 MB) Click below to read/download individual chapters. >> endobj Assume K1 ˙ K2 ˙ K3 ˙ form a decreasing sequence of closed subsets of X. Example 1. endobj Real Variables with Basic Metric Space Topology This is a text in elementary real analysis. Exercises) ��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$6`�i&�/q*MN5fE� ��o80}�;��Z%�ن��+6�lp}5����ut��ζ�����tu��`���l����q��j0�]�����q�Jh�P���������D���b�L�y��B�"��h�Kcghbu�1p�2q,��&��Xqp��-���U�t�j���B��X8 ʋ�5�T�@�4K @�D�~�VI�h�);4nc��:��B)������ƫ��3蔁�
�[)�_�ָGa�k�-Z0�U����[ڄ�'�;v��ѧ��:��d��^��gU#!��ң�� hޔX�n��}�W�L�\��M��$@�� << 24 0 obj << Real Variables with Basic Metric Space Topology. 99 0 obj 13 0 obj /Type /Annot Suppose {x n} is a convergent sequence which converges to two diﬀerent limits x 6= y. Proof. 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 . 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. /Rect [154.959 238.151 236.475 247.657] /Type /Annot /Rect [154.959 439.268 286.011 450.895] When dealing with an arbitrary metric space there may not be some natural fixed point 0. 97 0 obj Example 1. /A << /S /GoTo /D (subsubsection.1.5.1) >> /Subtype /Link 49 0 obj [3] Completeness (but not completion). endobj /Rect [154.959 456.205 246.195 467.831] The most familiar is the real numbers with the usual absolute value. Completeness) << /S /GoTo /D (subsection.1.4) >> 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. (2.1.1. (1.4.1. endobj /Subtype /Link /Rect [154.959 170.405 236.475 179.911] 123 0 obj /Subtype /Link �+��˞�H�,|,�f�Z[�E�ZT/� P*ј
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�ƽW�e��W���>����ml� A metric space consists of a set Xtogether with a function d: X X!R such that: (1) For … (1.3. /Subtype /Link 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� endobj /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. A subset of a metric space inherits a metric. << /S /GoTo /D (section.2) >> 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. a metric space. /Type /Annot /Type /Annot /Border[0 0 0]/H/I/C[1 0 0] (1.5.1. << /S /GoTo /D (subsubsection.2.1.1) >> TO REAL ANALYSIS William F. Trench AndrewG. 2. /Subtype /Link Recall that a Banach space is a normed vector space that is complete in the metric associated with the norm. /A << /S /GoTo /D (subsection.2.1) >> $\begingroup$ Singletons sets are always closed in a Hausdorff space and it is easy to show that metric spaces are Hausdorff. 28 0 obj 53 0 obj << /S /GoTo /D (subsubsection.1.1.1) >> >> 101 0 obj Let be a metric space. The Metric space

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