>> /A << /S /GoTo /D (subsection.1.2) >> >> [3] Completeness (but not completion). Fourier analysis. /Subtype /Link /Rect [154.959 170.405 236.475 179.911] Spaces of Functions) If each Kn 6= ;, then T n Kn 6= ;. stream /Subtype /Link endobj Properties of open subsets and a bit of set theory16 3.3. endobj (1.5.1. 254 Appendix A. 1.2 Open and Closed Sets In this section we review some basic definitions and propositions in topology. is complete if it’s complete as a metric space, i.e., if all Cauchy sequences converge to elements of the n.v.s. 53 0 obj (1.6.1. Math 4317 : Real Analysis I Mid-Term Exam 1 25 September 2012 Instructions: Answer all of the problems. 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 > /Border[0 0 0]/H/I/C[1 0 0] Let XˆRn be compact and f: X!R be a continuous function. endobj Real Variables with Basic Metric Space Topology (78 MB) Click below to read/download individual chapters. When metric dis understood, we often simply refer to Mas the metric space. To show that X is endobj A subset of a metric space inherits a metric. 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 >> /Subtype /Link Sequences in R 11 §2.2. endobj �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� A subset is called -net if A metric space is called totally bounded if finite -net. /A << /S /GoTo /D (section.2) >> NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. /Rect [154.959 439.268 286.011 450.895] 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. /Type /Page Recall that every normed vector space is a metric space, with the metric d(x;x0) = kx x0k. 2 Arbitrary unions of open sets are open. endobj d(f,g) is not a metric in the given space. 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. The topology of metric spaces) These Completeness) endobj METRIC SPACES 5 Remark 1.1.5. /A << /S /GoTo /D (subsection.1.4) >> The most familiar is the real numbers with the usual absolute value. (1. << /S /GoTo /D (subsubsection.1.1.2) >> 103 0 obj 0 When dealing with an arbitrary metric space there may not be some natural fixed point 0. 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 /Rect [154.959 422.332 409.953 433.958] h�b```f``�c`e`��e`@ �+G��p3�� Exercises) Measure density from extension 75 9.2. Therefore our de nition of a complete metric space applies to normed vector spaces: an n.v.s. /Rect [154.959 252.967 438.101 264.593] << /Rect [154.959 354.586 327.326 366.212] << (1.1. ri��֍5O�~G�����aP�{���s3^�v/:0Y�y�ۆ�ԏ�̌�1�Uǭw�D It covers in detail the Meaning, Definition and Examples of Metric Space. Afterall, for a general topological space one could just nilly willy define some singleton sets as open. /Rect [154.959 322.834 236.475 332.339] ��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� /Border[0 0 0]/H/I/C[1 0 0] Continuity) 115 0 obj This is a text in elementary real analysis. >> ��kԩ��wW���ё��,���eZg��t]~��p�蓇�Qi����F�;�������� iK� << /S /GoTo /D [86 0 R /Fit] >> The abstract concepts of metric spaces are often perceived as difficult. The term real analysis is a little bit of a misnomer. Metric space 2 §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. << 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. $\begingroup$ Singletons sets are always closed in a Hausdorff space and it is easy to show that metric spaces are Hausdorff. Informally: the distance from to is zero if and only if and are the same point,; the distance between two distinct points is positive, /Rect [154.959 219.094 249.277 230.721] /Type /Annot << /A << /S /GoTo /D (subsubsection.1.5.1) >> 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. The limit of a sequence in a metric space is unique. The characterization of continuity in terms of the pre-image of open sets or closed sets. << << $\endgroup$ – Squirtle Oct 1 '15 at 3:50 /A << /S /GoTo /D (subsubsection.1.1.3) >> Sequences 11 §2.1. endobj Real Variables with Basic Metric Space Topology. endstream endobj 72 0 obj <>stream endobj Skip to content. /Rect [154.959 337.649 310.461 349.276] 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. ... analysis, that is, the reader ha s familiarity with concepts li ke convergence of sequence of . 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! Recall that a Banach space is a normed vector space that is complete in the metric associated with the norm. << /S /GoTo /D (subsubsection.1.1.1) >> 8 0 obj /Border[0 0 0]/H/I/C[1 0 0] p. cm. Exercises) A subset of the real numbers is bounded whenever all its elements are at most some fixed distance from 0. Metric Spaces, Topological Spaces, and Compactness Proposition A.6. 105 0 obj The “classical Banach spaces” are studied in our Real Analysis sequence (MATH /Type /Annot endobj 52 0 obj endobj Example 1. /Type /Annot endobj /Type /Annot (If the Banach space 28 0 obj 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 . /Type /Annot Moore Instructor at M.I.T., just two years after receiving his Ph.D. at Duke University in 1949. << /S /GoTo /D (subsubsection.1.2.1) >> 87 0 obj More xڕWKS�8��+t����zZ� P��1���ڂ9G�86c;���eɁ���Zw���%����� ��=�|9c 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. 9 0 obj 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. 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 The set of real numbers R with the function d(x;y) = jx yjis a metric space. Sequences in metric spaces 13 << ��*McL� Oz?�K��z��WE��2�+%4�Dp�n�yRTͺ��U P@���{ƕ�M�rEo���0����OӉ� 60 0 obj 2. << >> Includes bibliographical references and index. �M)I$����Qo_D� >> /A << /S /GoTo /D (section*.3) >> /Border[0 0 0]/H/I/C[1 0 0] endobj 17 0 obj endobj %���� Convergence of sequences in metric spaces23 4. /Subtype /Link 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), /Border[0 0 0]/H/I/C[1 0 0] Often, if the metric dis clear from context, we will simply denote the metric space (X;d) by Xitself. Proof. /Type /Annot >> /Filter /FlateDecode /A << /S /GoTo /D (subsubsection.2.1.1) >> /Border[0 0 0]/H/I/C[1 0 0] << (2.1. /Rect [154.959 456.205 246.195 467.831] Extension results for Sobolev spaces in the metric setting 74 9.1. 20 0 obj /Type /Annot /Border[0 0 0]/H/I/C[1 0 0] /A << /S /GoTo /D (subsubsection.1.4.1) >> endobj NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. /Border[0 0 0]/H/I/C[1 0 0] /Type /Annot /A << /S /GoTo /D (subsubsection.1.6.1) >> endobj Exercises) /Length 2458 He wrote the first of these while he was a C.L.E. Assume K1 ˙ K2 ˙ K3 ˙ form a decreasing sequence of closed subsets of X. ��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$6`�i&�/q*MN5fE� ��o80}�;��Z%�ن��+6�lp}5����ut��ζ�����tu��`���l����q��j0�]�����q�Jh�P���������D���b�L�y��B�"��h�Kcghbu�1p�2q,��&��Xqp��-���U�t�j���B��X8 ʋ�5�T�@�4K @�D�~�VI�h�);4nc��:��B)������ƫ��3蔁� �[)�_�ָGa�k�-Z0�U����[ڄ�'�;v��ѧ��:��d��^��gU#!��ң�� /Rect [154.959 151.348 269.618 162.975] Throughout this section, we let (X,d) be a metric space unless otherwise specified. /Subtype /Link << /Type /Annot The fact that every pair is "spread out" is why this metric is called discrete. ... we have included a section on metric space completion. endobj << TO REAL ANALYSIS William F. Trench AndrewG. Some general notions A basic scenario is that of a measure space (X,A,µ), stream Real Variables with Basic Metric Space Topology This is a text in elementary real analysis. endstream endobj startxref ��1I�|����Y�=�� -a�P�#�L\�|'m6�����!K�zDR?�Uڭ�=��->�5�Fa�@��Y�|���W�70 /Rect [154.959 119.596 236.475 129.102] h�bbd``b`��@�� H��<3@�P ��b� �: ��H�u�ĜA괁�+��^$��AJN��ɲ����AF�1012\�10,���3� lw /Type /Annot Definition. Dense sets of continuous functions and the Stone-Weierstrass theorem) endobj >> /Rect [154.959 204.278 236.475 213.784] 80 0 obj << /S /GoTo /D (subsubsection.1.1.3) >> endobj << 1. /Subtype /Link /Rect [154.959 405.395 329.615 417.022] I prefer to use simply analysis. endobj >> << Notes (not part of the course) 10 Chapter 2. 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� metric space is call ed the 2-dimensional Euclidean Space . 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. Exercises) 44 0 obj /Border[0 0 0]/H/I/C[1 0 0] Example 4 .4 Taxi Cab Metric on Let be the set of all ordered pairs of real numbers and be a function (1.6. << /S /GoTo /D (subsection.1.5) >> Table of Contents << /Subtype /Link 94 7. This means that ∅is open in X. 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 3.A sequence fs ngconverges to sif and only if every subsequence fs n k gconverges to s. View Notes - metric_spaces.pdf from MATH 407 at University of Maryland, Baltimore County. De nitions, and open sets. WORKSHEET FOR THE PRELIMINARY EXAMINATION-REAL ANALYSIS (GENERAL TOPOLOGY, METRIC SPACES AND CONTINUITY)3 Problem 14. 57 0 obj 36 0 obj 84 0 obj 92 0 obj To show that (X;d) is indeed a metric space is left as an exercise. 89 0 obj 29 0 obj /A << /S /GoTo /D (subsection.1.6) >> endobj 90 0 obj PDF files can be viewed with the free program Adobe Acrobat Reader. Then this does define a metric, in which no distinct pair of points are "close". %%EOF /MediaBox [0 0 612 792] (1.2.2. 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 /Border[0 0 0]/H/I/C[1 0 0] Metric spaces definition, convergence, examples) Throughout this section, we let (X,d) be a metric space unless otherwise specified. 98 0 obj endobj 24 0 obj /Type /Annot /Subtype /Link >> For instance: << Contents Preface vii Chapter 1. The ℓ 0-normed space is studied in functional analysis, probability theory, and harmonic analysis. Distance in R 2 §1.2. ��WG�!����Є�+O8�ǚ�Sk���byߗ��1�F��i��W-$�N�s���;�ؠ��#��}�S��î6����A�iOg���V�u�xW����59��i=2̛�Ci[�m��(�]�tG��ށ馤W��!Q;R�͵�ә0VMN~���k�:�|*-����ye�[m��a�T!,-s��L�� (1.3. Metric Spaces, Topological Spaces, and Compactness Proposition A.6. 69 0 obj 32 0 obj /Type /Annot This book offers a unique approach to the subject which gives readers the advantage of a new perspective on ideas familiar from the analysis of a real line. Real Analysis (MA203) AmolSasane. endobj /Border[0 0 0]/H/I/C[1 0 0] 102 0 obj /Subtype /Link %PDF-1.5 Neighbourhoods and open sets 6 §1.4. Compactness) endobj For the purposes of boundedness it does not matter. /Subtype /Link /Border[0 0 0]/H/I/C[1 0 0] Real Variables with Basic Metric Space Topology. 0�M�������ϊM���D��"����́_~.pX8�^8�ZGxd0����?�������;ݦ��?�K-H�E��73�s��#b��Wkv�5]��*d����m?ll{i�O!��(�c�.Aԧ�*l�Y$��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�[� �܉d`g��e�@ދ�͢&�k���͕��Ue��[�-�-B��S�cdF�&c�K��i�l�WdyOF�-Ͷ�n^]~ Normed real vector spaces9 2.2. /A << /S /GoTo /D (subsubsection.1.1.2) >> endobj endobj (1.3.1. Real Variables with Basic Metric Space Topology This is a text in elementary real analysis. 123 0 obj 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. /Subtype /Link >> 40 0 obj << /S /GoTo /D (section.1) >> << /S /GoTo /D (subsection.1.2) >> The real valued function f is continuous at a Å R , iff whenever { :J } á @ 5 is the << /S /GoTo /D (subsection.1.3) >> /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 >> << /S /GoTo /D (subsection.2.1) >> << Exercises) Neighbourhoods and open sets 6 §1.4. (1.1.1. endstream endobj 69 0 obj <> endobj 70 0 obj <> endobj 71 0 obj <>stream Click below to read/download the entire book in one pdf file. 68 0 obj Later endobj Real Analysis Muruhan Rathinam February 19, 2019 1 Metric spaces and sequences in metric spaces 1.1 Metric /A << /S /GoTo /D (subsubsection.1.2.2) >> Real Analysis MCQs 01 consist of 69 most repeated and most important questions. 12 0 obj endobj Lecture notes files. 4.1.3, Ex. 85 0 obj 94 0 obj R, metric spaces and Rn 1 §1.1. 1 If X is a metric space, then both ∅and X are open in X. /Type /Annot endobj >> endobj >> /Type /Annot /Subtype /Link View Notes - metric_spaces.pdf from MATH 407 at University of Maryland, Baltimore County. 111 0 obj Together with Y, the metric d Y defines the automatic metric space (Y,d Y). 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. Together with Y, the metric d Y defines the automatic metric space (Y,d Y). 72 0 obj 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`�� << /A << /S /GoTo /D (subsection.1.1) >> For the purposes of boundedness it does not matter. Topics covered includes: Upper and Lower Limits of Sequences of Real Numbers, Continuous Functions, Differentiation, Riemann-Stieltjes Integration, Unifom Convergence … 4 0 obj >> One can do more on a metric space. >> >> This section records notations for spaces of real functions. endobj /Subtype /Link arrive at metric spaces and prove Picard’s theorem using the fixed point theorem as is usual. /A << /S /GoTo /D (subsubsection.1.1.1) >> >> endobj endobj /Resources 108 0 R >> Let Xbe a compact metric space. Analysis on metric spaces 1.1. << 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! << /S /GoTo /D (section*.3) >> << 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 endobj >> >> A metric space consists of a set Xtogether with a function d: X X!R such that: (1) For … /Rect [154.959 373.643 236.475 383.149] 73 0 obj endobj << /Subtype /Link /A << /S /GoTo /D (subsection.1.3) >> PDF | This chapter will ... and metric spaces. Real analysis with real applications/Kenneth R. Davidson, Allan P. Donsig. endobj 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. endobj 86 0 obj /A << /S /GoTo /D (section*.2) >> endobj /Subtype /Link /Subtype /Link Similarly, Q with the Euclidean (absolute value) metric is also a metric space. Real Analysis: Part II William G. Faris June 3, 2004. ii. 41 0 obj << /S /GoTo /D (subsubsection.1.6.1) >> Contents Preface vii Chapter 1. Basics of Metric spaces) %PDF-1.5 %���� 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. << /S /GoTo /D (subsubsection.1.4.1) >> h��X�O�H�W�c� 93 0 obj Real Analysis Muruhan Rathinam February 19, 2019 1 Metric spaces and sequences in metric spaces 1.1 Metric 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. 16 0 obj /Rect [154.959 272.024 206.88 281.53] We can also define bounded sets in a metric space. The closure of a subset of a metric space. distance function in a metric space, we can extend these de nitions from normed vector spaces to general metric spaces. endobj Real Variables with Basic Metric Space Topology (78 MB) Click below to read/download individual chapters. Often, if the metric dis clear from context, we will simply denote the metric space (X;d) by Xitself. /Type /Annot 100 0 obj 68 0 obj <> endobj << k, is an example of a Banach space. (1.4.1. endobj endobj [prop:mslimisunique] A convergent sequence in a metric space … 109 0 obj 37 0 obj Let Xbe a compact metric space. endobj This is a text in elementary real analysis. << /S /GoTo /D (section.2) >> endstream Sequences 11 §2.1. /Border[0 0 0]/H/I/C[1 0 0] 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. Metric spaces: basic definitions5 2.1. (1.4. >> 56 0 obj << /S /GoTo /D (subsubsection.1.3.1) >> /Rect [154.959 136.532 517.072 146.038] /Border[0 0 0]/H/I/C[1 0 0] Example 1. << /S /GoTo /D (subsection.1.4) >> /Subtype /Link 101 0 obj endobj De nitions (2 points each) 1.State the de nition of a metric space. >> The space of sequences has a complete metric topology provided by the F-norm ↦ ∑ − | | + | |, which is discussed by Stefan Rolewicz in Metric Linear Spaces. Click below to read/download the entire book in one pdf file. a metric space. 45 0 obj /Border[0 0 0]/H/I/C[1 0 0] 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. Discussion of open and closed sets in subspaces. A metric space can be thought of as a very basic space having a geometry, with only a few axioms. 76 0 obj ISBN 0-13-041647-9 1. /A << /S /GoTo /D (subsubsection.1.3.1) >> /Border[0 0 0]/H/I/C[1 0 0] endobj /Contents 109 0 R Topics covered includes: Upper and Lower Limits of Sequences of Real Numbers, Continuous Functions, Differentiation, Riemann-Stieltjes Integration, Unifom Convergence … We must replace \(\left\lvert {x-y} \right\rvert\) with \(d(x,y)\) in the proofs and apply the triangle inequality correctly. /Type /Annot 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. 48 0 obj We review open sets, closed sets, norms, continuity, and closure. 77 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. /Rect [154.959 303.776 235.298 315.403] The closure of a subset of a metric space. ə�t�SNe���}�̅��l��ʅ$[���Ȑ8kd�C��eH�E[\���\��z��S� $O� (2. 13 0 obj 108 0 obj Example 1.7. (2.1.1. 254 Appendix A. 118 0 obj <>stream << /D [86 0 R /XYZ 315.372 499.67 null] /A << /S /GoTo /D (subsection.2.1) >> (References) Suppose {x n} is a convergent sequence which converges to two different limits x 6= y. (Acknowledgements) Chapter 1 Metric Spaces These notes accompany the Fall 2011 Introduction to Real Analysis course 1.1 De nition and Examples De nition 1.1. /Rect [154.959 185.221 246.864 196.848] /Type /Annot Continuous functions between metric spaces26 4.1. uN3���m�'�p��O�8�N�߬s�������;�a�1q�r�*��øs �F���ϛO?3�o;��>W�A�v<>U����zA6���^p)HBea�3��n숎�*�]9���I�f��v�j�d�翲4$ž.�,7��j��qg[?��&N���1E�蜭��*�����)ܻ)ݎ���.G�[�}xǨO�f�"h���|dx8w�s���܂ 3̢MA�G�Pَ]�6�"�EJ������ 1 if X is pdf | this Chapter will... and metric spaces, Topological spaces, Topological spaces Topological! That ( X ; d ) by Xitself for a general Topological space one could nilly! Is `` spread out '' is why this metric is called -net if a metric space, let... 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