/Subtype /Link %���� << (1.1. R, metric spaces and Rn 1 §1.1. /Rect [154.959 373.643 236.475 383.149] (2. (1.1.2. 7.1. endobj Later %%EOF Includes bibliographical references and index. Real Variables with Basic Metric Space Topology This is a text in elementary real analysis. /Border[0 0 0]/H/I/C[1 0 0] << Real Variables with Basic Metric Space Topology This is a text in elementary real analysis. 110 0 obj /Type /Annot k, is an example of a Banach space. 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. 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. endobj 60 0 obj Let $$(X,d)$$ be a metric space. /A << /S /GoTo /D (subsubsection.1.6.1) >> We review open sets, closed sets, norms, continuity, and closure. endobj 77 0 obj 115 0 obj For the purposes of boundedness it does not matter. Distance in R 2 §1.2. 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. 1 If X is a metric space, then both ∅and X are open in X. /Type /Annot << Suppose {x n} is a convergent sequence which converges to two diﬀerent limits x 6= y. The closure of a subset of a metric space. d(f,g) is not a metric in the given space. h�bbdb��@�� H��<3@�P ��b� �: ��H�u�ĜA괁�+��^$��AJN��ɲ����AF�1012\�10,���3� lw /Border[0 0 0]/H/I/C[1 0 0] 44 0 obj Skip to content. WORKSHEET FOR THE PRELIMINARY EXAMINATION-REAL ANALYSIS (GENERAL TOPOLOGY, METRIC SPACES AND CONTINUITY)3 Problem 14. Exercises) [prop:mslimisunique] A convergent sequence in a metric space … endobj << /S /GoTo /D (subsubsection.1.1.2) >> More In some contexts it is convenient to deal instead with complex functions; ... the metric space is itself a vector space in a natural way. /Subtype /Link endstream >> Spaces of Functions) 1. Metric spaces are generalizations of the real line, in which some of the theorems that hold for R remain valid. 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. Together with Y, the metric d Y deﬁnes the automatic metric space (Y,d Y). 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 Euclidean metric. /Rect [154.959 151.348 269.618 162.975] Topics covered includes: Upper and Lower Limits of Sequences of Real Numbers, Continuous Functions, Differentiation, Riemann-Stieltjes Integration, Unifom Convergence … 45 0 obj �+��˞�H�,޴|,�f�Z[�E�ZT/� P*ј � �ƽW�e��W���>����ml� /Border[0 0 0]/H/I/C[1 0 0] Assume K1 ˙ K2 ˙ K3 ˙ form a decreasing sequence of closed subsets of X. Proof. 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. Sequences in metric spaces 13 Proof. endobj The fact that every pair is "spread out" is why this metric is called discrete. 57 0 obj /Border[0 0 0]/H/I/C[1 0 0] /Subtype /Link >> endobj 102 0 obj �;ܻ�r��׹�g���b��B^�ʈ��/�!��4�9yd�HQ"�aɍ�Y�a�%���5���{z�-)B�O��(�د�];��%��� ݦ�. NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. 1.2 Open and Closed Sets In this section we review some basic deﬁnitions and propositions in topology. /A << /S /GoTo /D (subsection.1.3) >> << /Contents 109 0 R ��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�:$�V�?b��q����H��n� Theorem 9.6 (Metric space is a topological space) Let (X,d)be a metric space. 33 0 obj /Border[0 0 0]/H/I/C[1 0 0] endobj /A << /S /GoTo /D (section.1) >> 69 0 obj De nitions (2 points each) 1.State the de nition of a metric space. 90 0 obj 98 0 obj These (1. p. cm. >> endobj Compactness) De nitions, and open sets. Let XˆRn be compact and f: X!R be a continuous function. Measure density from extension 75 9.2. Chapter 1 Metric Spaces These notes accompany the Fall 2011 Introduction to Real Analysis course 1.1 De nition and Examples De nition 1.1. << /S /GoTo /D (subsubsection.2.1.1) >> (1.2.1. 72 0 obj ISBN 0-13-041647-9 1. metric space is call ed the 2-dimensional Euclidean Space . endstream endobj 69 0 obj <> endobj 70 0 obj <> endobj 71 0 obj <>stream /Border[0 0 0]/H/I/C[1 0 0] �B�L�N���=x���-qk������([��">��꜋=��U�yFѱ.,�^����seT���[��W�ECp����U�S��N�F������ �$ 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. /A << /S /GoTo /D (subsubsection.2.1.1) >> 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. Assume K1 ˙ K2 ˙ K3 ˙ form a decreasing sequence of closed subsets of X. << /S /GoTo /D (subsubsection.1.2.2) >> endobj 88 0 obj << /S /GoTo /D (subsection.1.6) >> stream Lec # Topics; 1: Metric Spaces, Continuity, Limit Points ()2: Compactness, Connectedness ()3: Differentiation in n Dimensions ()4: Conditions … 108 0 obj Metric Spaces (10 lectures) Basic de…nitions: metric spaces, isometries, continuous functions ( ¡ de…nition), homeo-morphisms, open sets, closed sets. R, metric spaces and Rn 1 §1.1. ��*McL� Oz?�K��z��WE��2�+%4�Dp�n�yRTͺ��U P@���{ƕ�M�rEo���0����OӉ� These are not the same thing. /A << /S /GoTo /D (subsubsection.1.1.1) >> 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$��8�ʗ�O1B�-K�����b�&����r���e�g�0�wV�X/��'2_������|v��٥uM�^��@v���1�m1��^Ύ/�U����c'e-���u�᭠��J�FD�Gl�R���_�0�/ 9/ [�x-�S�ז��/���4E9�Ս�����&�z���}�5;^N0ƺ�N����-)o�[� �܉dg��e�@ދ�͢&�k���͕��Ue��[�-�-B��S�cdF�&c�K��i�l�WdyOF�-Ͷ�n^]~ (2.1.1. I prefer to use simply analysis. Exercises) /D [86 0 R /XYZ 144 720 null] The real valued function f is continuous at a Å R , iff whenever { :J } á @ 5 is the (1.3. /A << /S /GoTo /D (subsection.1.2) >> Real Analysis Muruhan Rathinam February 19, 2019 1 Metric spaces and sequences in metric spaces 1.1 Metric Metric spaces: basic deﬁnitions5 2.1. We can also define bounded sets in a metric space. Some of the main results in real analysis are (i) Cauchy sequences converge, (ii) for continuous functions f(lim n!1x n) = lim n!1f(x n), /Type /Annot /Rect [154.959 252.967 438.101 264.593] xڕ˒�6��P�e�*�&� kkv�:�MbWœ��䀡 �e���1����(Q����h�F��갊V߽z{����$Z��0�Z��W*IVF�H���n�9��[U�Q|���Oo����4 ެ�"����?��i���^EB��;]�TQ!�t�u���@Q)�H��/M��S�vwr��#���TvM�� Metric Spaces Joseph Muscat2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) 94 0 obj It covers in detail the Meaning, Definition and Examples of Metric Space. endobj << /S /GoTo /D (section*.2) >> 21 0 obj /Rect [154.959 388.459 318.194 400.085] endstream endobj startxref endobj 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. Analysis on metric spaces 1.1. ri��֍5O�~G�����aP�{���s3^�v/:0Y�y�ۆ�ԏ�̌�1�Uǭw�D /Border[0 0 0]/H/I/C[1 0 0] 24 0 obj If each Kn 6= ;, then T n Kn 6= ;. >> /D [86 0 R /XYZ 143 742.918 null] 106 0 obj /Subtype /Link 97 0 obj 68 0 obj <> endobj Real Analysis MCQs 01 consist of 69 most repeated and most important questions. h��X�O�H�W�c� (1.3.1. /Border[0 0 0]/H/I/C[1 0 0] Contents Preface vii Chapter 1. hޔX�n��}�W�L�\��M��$@�� /Length 2458 /ProcSet [ /PDF /Text ] endobj 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 << >> In a complete metric space Every sequence converges Every cauchy sequence converges there is … endobj endobj Basics of Metric spaces) (1.1.1. 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! Example 4 .4 Taxi Cab Metric on Let be the set of all ordered pairs of real numbers and be a function endobj (1.4. 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 For example, R3 is a metric space when we consider it together with the Euclidean distance. Spaces is a modern introduction to real analysis at the advanced undergraduate level. >> endobj /Rect [154.959 456.205 246.195 467.831] TO REAL ANALYSIS William F. Trench AndrewG. << /Subtype /Link Metric Spaces, Topological Spaces, and Compactness Proposition A.6. endobj Proof. stream endobj Exercises) Given a set X a metric on X is a function d: X X!R endobj Notes (not part of the course) 10 Chapter 2. >> ��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�� A metric space consists of a set Xtogether with a function d: X X!R such that: (1) For … /Subtype /Link In other words, no sequence may converge to two diﬀerent limits. 0 endobj A subset of a metric space inherits a metric. << /S /GoTo /D (subsubsection.1.3.1) >> 49 0 obj 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. The other type of analysis, complex analysis, really builds up on the present material, rather than being distinct. Metric Spaces, Topological Spaces, and Compactness Proposition A.6. endobj A metric space can be thought of as a very basic space having a geometry, with only a few axioms. 103 0 obj The discrete metric on the X is given by : d(x, y) = 0 if x = y and d(x, y) = 1 otherwise. We must replace $$\left\lvert {x-y} \right\rvert$$ with $$d(x,y)$$ in the proofs and apply the triangle inequality correctly. Informally: the distance from to is zero if and only if and are the same point,; the distance between two distinct points is positive, << /S /GoTo /D [86 0 R /Fit] >> endobj 4.4.12, Def. >> /A << /S /GoTo /D (subsubsection.1.1.3) >> 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. /Rect [154.959 185.221 246.864 196.848] 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. << /S /GoTo /D (section.1) >> 65 0 obj /Rect [154.959 219.094 249.277 230.721] Example 1.7. 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. 254 Appendix A. Real Analysis (MA203) AmolSasane. NPTEL provides E-learning through online Web and Video courses various streams. << << endobj 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. Lecture notes files. ... we have included a section on metric space completion. >> [3] Completeness (but not completion). He wrote the first of these while he was a C.L.E. 101 0 obj /Rect [154.959 288.961 236.475 298.466] Introduction Let X be an arbitrary set, which could consist of vectors in Rn, functions, sequences, matrices, etc. $\endgroup$ – Squirtle Oct 1 '15 at 3:50 >> /Subtype /Link << 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. 13 0 obj Often, if the metric dis clear from context, we will simply denote the metric space (X;d) by Xitself. /Rect [154.959 238.151 236.475 247.657] Real Variables with Basic Metric Space Topology (78 MB) Click below to read/download individual chapters. The space of sequences has a complete metric topology provided by the F-norm ↦ ∑ − | | + | |, which is discussed by Stefan Rolewicz in Metric Linear Spaces. /A << /S /GoTo /D (section.2) >> (1.4.1. >> 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 . View Notes - metric_spaces.pdf from MATH 407 at University of Maryland, Baltimore County. /A << /S /GoTo /D (section*.2) >> >> Metric spaces definition, convergence, examples) Contents Preface vii Chapter 1. (1.6.1. /Rect [154.959 303.776 235.298 315.403] Discussion of open and closed sets in subspaces. Topics covered includes: Upper and Lower Limits of Sequences of Real Numbers, Continuous Functions, Differentiation, Riemann-Stieltjes Integration, Unifom Convergence … /Border[0 0 0]/H/I/C[1 0 0] << /S /GoTo /D (subsection.2.1) >> (1.6. 36 0 obj /A << /S /GoTo /D (subsubsection.1.4.1) >> 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. << /S /GoTo /D (section*.3) >> /A << /S /GoTo /D (section*.3) >> The term real analysis is a little bit of a misnomer. 96 0 obj 52 0 obj /Type /Annot (References) /Parent 120 0 R Some general notions A basic scenario is that of a measure space (X,A,µ), (1.5. The second is the set that contains the terms of the sequence, and if /Type /Annot Exercises) 123 0 obj << << endobj 94 7. 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. Let X be a metric space. Dense sets of continuous functions and the Stone-Weierstrass theorem) /Border[0 0 0]/H/I/C[1 0 0] Sequences in R 11 §2.2. �s /Type /Annot So for each vector endobj endobj << /S /GoTo /D (subsection.1.1) >> << /S /GoTo /D (subsection.1.3) >> So prepare real analysis to attempt these questions. /Rect [154.959 405.395 329.615 417.022] endobj Throughout this section, we let (X,d) be a metric space unless otherwise speciﬁed. Exercises) 5.1.1 and Theorem 5.1.31. Real analysis with real applications/Kenneth R. Davidson, Allan P. Donsig. Sequences in R 11 §2.2. Neighbourhoods and open sets 6 §1.4. << /Subtype /Link /Type /Annot Why the triangle inequality?) /MediaBox [0 0 612 792] 48 0 obj %PDF-1.5 %���� 53 0 obj The “classical Banach spaces” are studied in our Real Analysis sequence (MATH For the purposes of boundedness it does not matter. 12 0 obj /Border[0 0 0]/H/I/C[1 0 0] 40 0 obj A subset of a metric space inherits a metric. 100 0 obj 89 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 … PDF files can be viewed with the free program Adobe Acrobat Reader. >> The ℓ 0-normed space is studied in functional analysis, probability theory, and harmonic analysis. �@� �YZ<5�e��SE� оs�~fx�u���� �Au�%���D]�,�Q�5�j�3���\�#�l��˖L�?�;8�5�@�{R[VS=���� /Subtype /Link 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. 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. 118 0 obj <>stream /A << /S /GoTo /D (subsection.2.1) >> %PDF-1.5 Real Analysis (MA203) AmolSasane. /A << /S /GoTo /D (subsection.1.1) >> 4.1.3, Ex. Fourier analysis. 254 Appendix A. 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. 86 0 obj This allows a treatment of Lp spaces as complete spaces of bona ﬁde functions, by 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! ə�t�SNe���}�̅��l��ʅ$[���Ȑ8kd�C��eH�E[\���\��z��S�$O� >> 91 0 obj << /S /GoTo /D (subsection.1.4) >> 9 0 obj Distance in R 2 §1.2. Example 7.4. To show that X is /Rect [154.959 337.649 310.461 349.276] /Filter /FlateDecode >> /Subtype /Link 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. Let be a metric space. Open subsets12 3.1. Extension from measure density 79 References 84 1. 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Space Topology ( 78 MB ) click below to read/download individual chapters absolute value willy define some singleton as...
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