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/Type /Annot %PDF-1.5 %�� The abstract concepts of metric spaces are often perceived as difficult. endobj /Type /Annot Spaces is a modern introduction to real analysis at the advanced undergraduate level. >> Recall that every normed vector space is a metric space, with the metric d(x;x0) = kx x0k. /Contents 109 0 R Then this does define a metric, in which no distinct pair of points are "close". 81 0 obj Continuous functions between metric spaces26 4.1. 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. Together with Y, the metric d Y deﬁnes the automatic metric space (Y,d Y). More Table of Contents 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. Closure, interior, density) Given a set X a metric on X is a function d: X X!R �+��˞�H�,޴|,�f�Z[�E�ZT/� P*ј � �ƽW�e��W��>��ml� /Type /Page /Subtype /Link Dense sets of continuous functions and the Stone-Weierstrass theorem) 104 0 obj endobj 20 0 obj 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. << /S /GoTo /D (section.1) >> Theorem 9.6 (Metric space is a topological space) Let (X,d)be a metric space. endobj /Subtype /Link endobj endobj h�b```f``�c`e`��e`@ �+G��p3�� << When metric dis understood, we often simply refer to Mas the metric space. >> /Type /Annot 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. 61 0 obj endobj For the purposes of boundedness it does not matter. /Border[0 0 0]/H/I/C[1 0 0] �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 endobj /Rect [154.959 119.596 236.475 129.102] ��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$6`�i&�/q*MN5fE� ��o80}�;��Z%�ن��+6�lp}5��ut��ζ��tu��`��l��q��j0�]��q�Jh�P��D��b�L�y��B�"��h�Kcghbu�1p�2q,��&��Xqp��-��U�t�j��B��X8 ʋ�5�T�@�4K @�D�~�VI�h�);4nc��:��B)��ƫ��3蔁� �[)�_�ָGa�k�-Z0�U��[ڄ�'�;v��ѧ��:��d��^��gU#!��ң�� endstream endobj startxref The space of sequences has a complete metric topology provided by the F-norm ↦ ∑ − | | + | |, which is discussed by Stefan Rolewicz in Metric Linear Spaces. /Rect [154.959 151.348 269.618 162.975] 103 0 obj << /S /GoTo /D (subsection.1.5) >> is complete if it’s complete as a metric space, i.e., if all Cauchy sequences converge to elements of the n.v.s. stream << 101 0 obj endobj Sequences in R 11 §2.2. Sequences 11 §2.1. The set of real numbers R with the function d(x;y) = jx yjis a metric space. h��X�O�H�W�c� Exercises) /Type /Annot WORKSHEET FOR THE PRELIMINARY EXAMINATION-REAL ANALYSIS (GENERAL TOPOLOGY, METRIC SPACES AND CONTINUITY)3 Problem 14. Recall that saying that (M,d(x,y))is a met-ric space means that Mis a nonempty set; d(x,y) is a function on M×Mtaking values in the non-negative real numbers; d(x,y)= 0if and only if ə�t�SNe��}�̅��l��ʅ$[��Ȑ8kd�C��eH�E[\��\��z��S� $O� A metric space can be thought of as a very basic space having a geometry, with only a few axioms. He wrote the first of these while he was a C.L.E. The function d is called the metric on X.It is also sometimes called a distance function or simply a distance.. Often d is omitted and one just writes X for a metric space if it is clear from the context what metric is being used.. We already know a few examples of metric spaces. >> << /S /GoTo /D (subsection.2.1) >> The closure of a subset of a metric space. 111 0 obj In the following we shall need the concept of the dual space of a Banach space E. The dual space E consists of all continuous linear functions from the Banach space to the real numbers. endobj 4.1.3, Ex. endobj Analysis on metric spaces 1.1. << This section records notations for spaces of real functions. Discussion of open and closed sets in subspaces. /Parent 120 0 R In the exercises you will see that the case m= 3 proves the triangle inequality for the spherical metric of Example 1.6. Neighbourhoods and open sets 6 §1.4. 89 0 obj /Rect [154.959 136.532 517.072 146.038] /Type /Annot /Subtype /Link Real Variables with Basic Metric Space Topology (78 MB) Click below to read/download individual chapters. endobj endobj (1.5.1. Proof. (1.3.1. /Type /Annot 2 Arbitrary unions of open sets are open. The ℓ 0-normed space is studied in functional analysis, probability theory, and harmonic analysis. /A << /S /GoTo /D (subsubsection.1.5.1) >> Metric space 2 §1.3. /A << /S /GoTo /D (subsection.1.4) >> >> << /S /GoTo /D (subsubsection.2.1.1) >> Throughout this section, we let (X,d) be a metric space unless otherwise speciﬁed. /Border[0 0 0]/H/I/C[1 0 0] Distance in R 2 §1.2. ��WG�!��Є�+O8�ǚ�Sk��byߗ��1�F��i��W-$�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�� (If the Banach space Notes (not part of the course) 10 Chapter 2. The characterization of continuity in terms of the pre-image of open sets or closed sets. ISBN 0-13-041647-9 1. << /Type /Annot uIM�ᓪlM ɳ\%� ��D��V��#\)��PB��\�ţY��v��~+�ېJ��Z��##�|]!�@�9>N�� /Border[0 0 0]/H/I/C[1 0 0] << /S /GoTo /D (subsubsection.1.5.1) >> The closure of a subset of a metric space. NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. /A << /S /GoTo /D (subsection.1.2) >> /Subtype /Link endobj << 100 0 obj 1 If X is a metric space, then both ∅and X are open in X. /Border[0 0 0]/H/I/C[1 0 0] The most familiar is the real numbers with the usual absolute value. << /A << /S /GoTo /D (subsubsection.1.3.1) >> 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 48 0 obj >> Euclidean metric. 90 0 obj 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 Continuity) Informally: the distance from to is zero if and only if and are the same point,; the distance between two distinct points is positive, Sequences 11 §2.1. >> De nitions, and open sets. /Rect [154.959 456.205 246.195 467.831] Spaces of Functions) << [3] Completeness (but not completion). Exercises) We can also define bounded sets in a metric space. << The second is the set that contains the terms of the sequence, and if 95 0 obj [3] Completeness (but not completion). p. cm. endobj << Example 7.4. /Rect [154.959 322.834 236.475 332.339] /Subtype /Link Introduction Let X be an arbitrary set, which could consist of vectors in Rn, functions, sequences, matrices, etc. >> �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� Lec # Topics; 1: Metric Spaces, Continuity, Limit Points ()2: Compactness, Connectedness ()3: Differentiation in n Dimensions ()4: Conditions … /Type /Annot /Border[0 0 0]/H/I/C[1 0 0] (1.6. endobj /Type /Annot /Type /Annot << >> 13 0 obj The fact that every pair is "spread out" is why this metric is called discrete. << /S /GoTo /D (subsubsection.1.1.2) >> endobj Real Variables with Basic Metric Space Topology This is a text in elementary real analysis. Real Variables with Basic Metric Space Topology. 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. The term real analysis is a little bit of a misnomer. (1.1.1. View Notes - metric_spaces.pdf from MATH 407 at University of Maryland, Baltimore County. endobj For example, R3 is a metric space when we consider it together with the Euclidean distance. 36 0 obj 69 0 obj endobj 9 0 obj /A << /S /GoTo /D (subsubsection.1.6.1) >> << endobj $\endgroup$ – Squirtle Oct 1 '15 at 3:50 Open subsets12 3.1. endobj endobj 16 0 obj /A << /S /GoTo /D (subsubsection.1.2.2) >> 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. Real Analysis MCQs 01 consist of 69 most repeated and most important questions. << ... we have included a section on metric space completion. /Type /Annot Many metric spaces are minor variations of the familiar real line. �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� /Subtype /Link << /S /GoTo /D (section.2) >> 123 0 obj endobj /A << /S /GoTo /D (section*.3) >> (1.1. endobj endobj /Subtype /Link /A << /S /GoTo /D (subsubsection.2.1.1) >> endobj >> >> >> Real Analysis (MA203) AmolSasane. A ball B of radius r around a point x ∈ X is B = {y ∈ X|d(x,y) < r}. endobj $\begingroup$ Singletons sets are always closed in a Hausdorff space and it is easy to show that metric spaces are Hausdorff. ... analysis, that is, the reader ha s familiarity with concepts li ke convergence of sequence of . /A << /S /GoTo /D (subsubsection.1.4.1) >> >> 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. (1.2.2. endstream h�bbd``b`��@�� H��<3@�P ��b� �: ��H�u�ĜA괁�+��^$��AJN��ɲ��AF�1012\�10,��3� lw �@� �YZ<5�e��SE� оs�~fx�u�� Au�%��D]�,�Q�5�j�3��\�#�l��˖L�?�;8�5�@�{R[VS=�� 85 0 obj 72 0 obj << /S /GoTo /D (subsection.1.1) >> 91 0 obj 88 0 obj Metric Spaces, Topological Spaces, and Compactness Proposition A.6. Let Xbe a compact metric space. 17 0 obj 8 0 obj 4 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! >> /Border[0 0 0]/H/I/C[1 0 0] /A << /S /GoTo /D (subsubsection.1.2.1) >> In other words, no sequence may converge to two diﬀerent limits. 86 0 obj 45 0 obj endobj Product spaces10 3. Neighbourhoods and open sets 6 §1.4. >> endobj << �B�`L�N��=x��-qk��([��">��꜋=��U�yFѱ.,�^�`��seT��[��W�ECp��U�S��N�F�� $ R, metric spaces and Rn 1 §1.1. 107 0 obj Assume K1 ˙ K2 ˙ K3 ˙ form a decreasing sequence of closed subsets of X. endobj Let $(X,d)$ be a metric space. 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! These Real Analysis Muruhan Rathinam February 19, 2019 1 Metric spaces and sequences in metric spaces 1.1 Metric Includes bibliographical references and index. The limit of a sequence of points in a metric space. Examples of metric spaces) endobj /Type /Annot /Filter /FlateDecode Let XˆRn be compact and f: X!R be a continuous function. /Subtype /Link �;ܻ�r��׹�g��b`��B^�ʈ��/�!��4�9yd�HQ"�aɍ�Y�a�%��5�`��{z�-)B�O��(�د�];��%�� ݦ�. ��kԩ��wW��ё��,��eZg��t]~��p�蓇�Qi��F�;�� iK� d(f,g) is not a metric in the given 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] Moore Instructor at M.I.T., just two years after receiving his Ph.D. at Duke University in 1949. endobj Later /Filter /FlateDecode To show that (X;d) is indeed a metric space is left as an exercise. << /S /GoTo /D (subsubsection.1.6.1) >> /D [86 0 R /XYZ 143 742.918 null] Together with Y, the metric d Y deﬁnes the automatic metric space (Y,d Y). 1 0 obj endobj De nitions (2 points each) 1.State the de nition of a metric space. << /S /GoTo /D (section*.2) >> /Subtype /Link Topics covered includes: Upper and Lower Limits of Sequences of Real Numbers, Continuous Functions, Differentiation, Riemann-Stieltjes Integration, Unifom Convergence … Metric Spaces, Topological Spaces, and Compactness Proposition A.6. Exercises) endobj Equivalent metrics13 3.2. /Subtype /Link /Subtype /Link (1.6.1. (Acknowledgements) endobj endobj Notes (not part of the course) 10 Chapter 2. (2.1.1. TO REAL ANALYSIS William F. Trench AndrewG. About the metric setting 72 9. 12 0 obj endobj << /S /GoTo /D (subsubsection.1.3.1) >> << /S /GoTo /D (subsubsection.1.1.3) >> Metric Spaces Joseph Muscat2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) endobj << /Border[0 0 0]/H/I/C[1 0 0] /Type /Annot 1. Contents Preface vii Chapter 1. 87 0 obj Example 1. Metric spaces definition, convergence, examples) endobj Math 4317 : Real Analysis I Mid-Term Exam 1 25 September 2012 Instructions: Answer all of the problems. ��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� 1.2 Open and Closed Sets In this section we review some basic deﬁnitions and propositions in topology. 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. >> We review open sets, closed sets, norms, continuity, and closure. Suppose {x n} is a convergent sequence which converges to two diﬀerent limits x 6= y. PDF files can be viewed with the free program Adobe Acrobat Reader. We review open sets, closed sets, norms, continuity, and closure. Metric spaces: basic deﬁnitions5 2.1. 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. /MediaBox [0 0 612 792] metric space is call ed the 2-dimensional Euclidean Space . /A << /S /GoTo /D (subsection.1.6) >> >> 109 0 obj /Subtype /Link %PDF-1.5 (2. Proof. If each Kn 6= ;, then T n Kn 6= ;. R, metric spaces and Rn 1 §1.1. 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. 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� The real valued function f is continuous at a Å R , iff whenever { :J } á @ 5 is the A metric space is a set Xtogether with a metric don it, and we will use the notation (X;d) for a metric space. /Rect [154.959 238.151 236.475 247.657] Contents Preface vii Chapter 1. 37 0 obj hޔX�n��}�W�L�\��M��$@�� Example 1.7. 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. oG}�{�hN�8��~�t��9��@. 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. (References) 32 0 obj endobj When dealing with an arbitrary metric space there may not be some natural fixed point 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. A subset of the real numbers is bounded whenever all its elements are at most some fixed distance from 0. 94 0 obj �s 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 . 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. Let $(X,d)$ be a metric space. 60 0 obj Definition. �M)I$��Qo_D� /Border[0 0 0]/H/I/C[1 0 0] 1.2 Open Sets (in a metric space) Now that we have a notion of distance, we can deﬁne what it means to be an open set in a metric space. 0�M��ϊM��D��"��́_~.pX8�^8�ZGxd0��?��;ݦ��?�K-H�E��73�s��#b��Wkv�5]��*d��m?ll{i�O!��(�c�.Aԧ�*l�Y$��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^]~ /Border[0 0 0]/H/I/C[1 0 0] The “classical Banach spaces” are studied in our Real Analysis sequence (MATH << 44 0 obj %%EOF /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. (2.1. << /S /GoTo /D [86 0 R /Fit] >> /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 ] NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. (1.5. Similarly, Q with the Euclidean (absolute value) metric is also a metric space. Real Analysis Muruhan Rathinam February 19, 2019 1 Metric spaces and sequences in metric spaces 1.1 Metric Properties of open subsets and a bit of set theory16 3.3. << The set of real numbers R with the function d(x;y) = jx yjis a metric space. Lecture notes files. endobj endobj << /S /GoTo /D (subsection.1.6) >> 97 0 obj << endobj endobj Real Analysis: Part II William G. Faris June 3, 2004. ii. endobj 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. 68 0 obj 41 0 obj 92 0 obj 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. Example 1. The Metric space > 0 53 0 obj Extension from measure density 79 References 84 1. /Rect [154.959 388.459 318.194 400.085] << Basics of Metric spaces) 254 Appendix A. << /S /GoTo /D (subsubsection.1.4.1) >> >> endobj endobj 1.2 Open and Closed Sets In this section we review some basic deﬁnitions and propositions in topology. A subset is called -net if A metric space is called totally bounded if finite -net. /Rect [154.959 405.395 329.615 417.022] 80 0 obj endstream endobj 69 0 obj <> endobj 70 0 obj <> endobj 71 0 obj <>stream << Table of Contents If each Kn 6= ;, then T n Kn 6= ;. 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 << /S /GoTo /D (subsubsection.1.1.1) >> [prop:mslimisunique] A convergent sequence in a metric space … /A << /S /GoTo /D (subsection.1.5) >> 52 0 obj Compactness) This means that ∅is open in X. Proof. Skip to content. << /S /GoTo /D (subsubsection.1.2.2) >> /Rect [154.959 373.643 236.475 383.149] /A << /S /GoTo /D (subsection.1.3) >> 5 0 obj >> Sequences in R 11 §2.2. In some contexts it is convenient to deal instead with complex functions; ... the metric space is itself a vector space in a natural way. << /S /GoTo /D (subsubsection.1.2.1) >> >> endobj endobj endobj << /S /GoTo /D (subsubsection.1.1.1) >> /A << /S /GoTo /D (section.1) >> This allows a treatment of Lp spaces as complete spaces of bona ﬁde functions, by Proof. Recall that a Banach space is a normed vector space that is complete in the metric associated with the norm. 93 0 obj endobj /Type /Annot /A << /S /GoTo /D (subsubsection.1.1.2) >> /Rect [154.959 252.967 438.101 264.593] /Subtype /Link Let be a metric space. 64 0 obj endobj /Border[0 0 0]/H/I/C[1 0 0] 21 0 obj Why the triangle inequality?) (1.3. Let X be a metric space. Other continuities and spaces of continuous functions) 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. >> /Border[0 0 0]/H/I/C[1 0 0] /A << /S /GoTo /D (section*.2) >> /Length 2458 a metric space. (1. << I prefer to use simply analysis. The discrete metric on the X is given by : d(x, y) = 0 if x = y and d(x, y) = 1 otherwise. 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. To show that X is << 25 0 obj 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. Introduction Let X be an arbitrary set, which could consist of vectors in Rn, functions, sequences, matrices, etc. Often, if the metric dis clear from context, we will simply denote the metric space (X;d) by Xitself. 98 0 obj Completeness) Fourier analysis. 7.1. /Type /Annot /Subtype /Link ri��֍5O�~G��aP�{��s3^�v/:0Y�y�ۆ�ԏ�̌�1�Uǭw�D stream (1.1.2. Click below to read/download the entire book in one pdf file. Measure density from extension 75 9.2. << 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$.�,7��j��qg[?��&N��1E�蜭��*��)ܻ)ݎ��.G�[�}xǨO�f�"h��|dx8w�s��܂ 3̢MA�G�Pَ]�6�"�EJ�� 254 Appendix A. /A << /S /GoTo /D (subsubsection.1.1.3) >> Topics covered includes: Upper and Lower Limits of Sequences of Real Numbers, Continuous Functions, Differentiation, Riemann-Stieltjes Integration, Unifom Convergence … It covers in detail the Meaning, Definition and Examples of Metric Space. Normed real vector spaces9 2.2. >> 84 0 obj >> /A << /S /GoTo /D (subsection.1.1) >> /Type /Annot (1.2. /Rect [154.959 354.586 327.326 366.212] On few occasions, I have also shown that if we want to extend the result from metric spaces to topological spaces, what kind of extra conditions need to be imposed on the topological space. endobj Metric Spaces (10 lectures) Basic de…nitions: metric spaces, isometries, continuous functions ( ¡ de…nition), homeo-morphisms, open sets, closed sets. 29 0 obj Exercises) 49 0 obj Throughout this section, we let (X,d) be a metric space unless otherwise speciﬁed. << /S /GoTo /D (section*.3) >> ��T!QҤi��H�z��&q!R^J\ ��qb��;��8�}��济J'^'W�DZE�hӄ1 _C��8K��8c4(%�3 �� Z Z��J"��U�"�K�&Bj$�1 ,�L��H %�(lk�Y1`�(�k1A�!�2ff�(?�D3�d��۷��|0��z0b�0%�ggQ�̡n-��L��* 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 other type of analysis, complex analysis, really builds up on the present material, rather than being distinct. Metric spaces are generalizations of the real line, in which some of the theorems that hold for R remain valid. PDF | This chapter will ... and metric spaces. 106 0 obj k, is an example of a Banach space. So for each vector 96 0 obj METRIC SPACES 5 Remark 1.1.5. We must replace $\left\lvert {x-y} \right\rvert$ with $d(x,y)$ in the proofs and apply the triangle inequality correctly. /Border[0 0 0]/H/I/C[1 0 0] endobj /Rect [154.959 204.278 236.475 213.784] Real Variables with Basic Metric Space Topology (78 MB) Click below to read/download individual chapters. << /S /GoTo /D (subsection.1.2) >> >> Exercises) For the purposes of boundedness it does not matter. True or False (1 point each) 1.The set Rn with the usual metric is a complete metric space. 108 0 obj 73 0 obj 76 0 obj ��s괷��2N��5��q��w�f��a髩F�e�z& Nr\��R�so+w��?e$�l�F�VqI՟��z��y�/�x� �r�/�40�u@ �p ��@0E@e�(B� D�z H�10�5i V ��OZ�UG!V !�s�wZ*00��dZ�q�� R7�[fF)�`�Hb^�nQ��R��pPb`��U݆�Y �sr� Exercises) Cowles Distinguished Professor Emeritus Departmentof Mathematics Trinity University San Antonio, Texas, USA ... Chapter 8 Metric Spaces 518 8.1 Introduction to Metric Spaces 518 8.2 Compact Sets in a Metric Space 535 8.3 Continuous Functions on Metric Spaces … endobj 105 0 obj /Border[0 0 0]/H/I/C[1 0 0] Often, if the metric dis clear from context, we will simply denote the metric space (X;d) by Xitself. 40 0 obj 33 0 obj For functions from reals to reals: f : (c;d) !R, y is the limit of f at x 0 if for each ">0 there is a (") >0 such that 0 endobj endobj Compactness in Metric SpacesCompact sets in Banach spaces and Hilbert spacesHistory and motivationWeak convergenceFrom local to globalDirect Methods in Calculus of VariationsSequential compactnessApplications in metric spaces Equivalence of Compactness Theorem In metric space, a subset Kis compact if and only if it is sequentially compact. xڕ˒�6��P�e�*�&� kkv�:�MbWœ��䀡 �e��1��(Q��h�F��갊V߽z{��$Z��0�Z��W*IVF�H��n�9��[U�Q|��Oo��4 ެ�"��?��i��^EB��;]�TQ!�t�u��@Q)�H��/M��S�vwr��#��TvM`�� /Subtype /Link /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 >> Extension results for Sobolev spaces in the metric setting 74 9.1. /Rect [154.959 185.221 246.864 196.848] Deﬁnition 1.2.1. Assume K1 ˙ K2 ˙ K3 ˙ form a decreasing sequence of closed subsets of X. For instance: 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), 4.4.12, Def. Afterall, for a general topological space one could just nilly willy define some singleton sets as open. (X;d) is bounded if its image f(D) is a bounded set. The purpose of this deﬁnition for a sequence is to distinguish the sequence (x n) n2N 2XN from the set fx n 2Xjn2Ng X. First, we prove 1. Real analysis with real applications/Kenneth R. Davidson, Allan P. Donsig. The topology of metric spaces) Let Xbe a compact metric space. 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