Recall that a Banach space is a normed vector space that is complete in the metric associated with the norm. (X;d) is bounded if its image f(D) is a bounded set. Together with Y, the metric d Y deﬁnes the automatic metric space (Y,d Y). 65 0 obj Normed real vector spaces9 2.2. Moore Instructor at M.I.T., just two years after receiving his Ph.D. at Duke University in 1949. For example, R3 is a metric space when we consider it together with the Euclidean distance. See, for example, Def. >> Then this does define a metric, in which no distinct pair of points are "close". 1. Discussion of open and closed sets in subspaces. Real Variables with Basic Metric Space Topology (78 MB) Click below to read/download individual chapters. So for each vector << endobj PDF files can be viewed with the free program Adobe Acrobat Reader. Real analysis with real applications/Kenneth R. Davidson, Allan P. Donsig. /Type /Annot /Subtype /Link Proof. (1. >> >> Exercises) View Notes - metric_spaces.pdf from MATH 407 at University of Maryland, Baltimore County. << endobj 24 0 obj ��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� He wrote the first of these while he was a C.L.E. >> endobj /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. These (1.5. Metric spaces definition, convergence, examples) 107 0 obj endobj /Rect [154.959 303.776 235.298 315.403] Example 1.7. /Subtype /Link << /S /GoTo /D (subsubsection.1.2.1) >> /Subtype /Link 84 0 obj endobj Why the triangle inequality?) $\begingroup$ Singletons sets are always closed in a Hausdorff space and it is easy to show that metric spaces are Hausdorff. Sequences in metric spaces 13 (Acknowledgements) Distance in R 2 §1.2. distance function in a metric space, we can extend these de nitions from normed vector spaces to general metric spaces. Measure density from extension 75 9.2. Together with Y, the metric d Y deﬁnes the automatic metric space (Y,d Y). /Type /Annot There is also analysis related to continuous functions, limits, compactness, and so forth, as on a topological space. /Length 2458 /Border[0 0 0]/H/I/C[1 0 0] Contents Preface vii Chapter 1. 87 0 obj >> Example 7.4. endobj When metric dis understood, we often simply refer to Mas the metric space. 1 Distance A metric space can be thought of as a very basic space having a geometry, with only a few axioms. /Border[0 0 0]/H/I/C[1 0 0] �;ܻ�r���g���b`��B^�ʈ��/�!��4�9yd�HQ"�aɍ�Y�a�%���5�`��{z�-)B�O��(�د�];��%���
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Introduction Let X be an arbitrary set, which could consist of vectors in Rn, functions, sequences, matrices, etc. The term real analysis is a little bit of a misnomer. /Rect [154.959 151.348 269.618 162.975] A metric space is a set Xtogether with a metric don it, and we will use the notation (X;d) for a metric space. One can do more on a metric space. endobj 81 0 obj 95 0 obj << /A << /S /GoTo /D (subsection.2.1) >> stream /Border[0 0 0]/H/I/C[1 0 0] endobj Proof. Let Xbe a compact metric space. ��h������;��[
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�[)�_�ָGa�k�-Z0�U����[ڄ�'�;v��ѧ��:��d��^��gU#!��ң�� 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 /Subtype /Link /Rect [154.959 354.586 327.326 366.212] endobj >> (1.5.1. >> ��T!QҤi��H�z��&q!R^J\ �����qb��;��8�}���济J'^'W�DZE�hӄ1
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%�(lk�Y1`�(�k1A�!�2ff�(?�D3�d����۷���|0��z0b�0%�ggQ�̡n-��L��* Definition. Open subsets12 3.1. xڕWKS�8��+t����zZ� P��1���ڂ9G�86c;���eɁ���Zw���%����� ��=�|9c Metric spaces are generalizations of the real line, in which some of the theorems that hold for R remain valid. Math 4317 : Real Analysis I Mid-Term Exam 1 25 September 2012 Instructions: Answer all of the problems. /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 >> 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. Closure, interior, density) /Type /Annot For the purposes of boundedness it does not matter. PDF files can be viewed with the free program Adobe Acrobat Reader. endobj Properties of open subsets and a bit of set theory16 3.3. << Let \((X,d)\) be a metric space. /Border[0 0 0]/H/I/C[1 0 0] endobj /Subtype /Link << >> /Type /Annot Lecture notes files. Skip to content. /MediaBox [0 0 612 792] Table of Contents �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� /Rect [154.959 322.834 236.475 332.339] /Type /Annot 44 0 obj /Subtype /Link The second is the set that contains the terms of the sequence, and if Exercises) /Subtype /Link h�bbd``b`��@�� H��<3@�P ��b� �: ��H�u�ĜA괁�+��^$��AJN��ɲ����AF�1012\�10,���3� lw
��1I�|����Y�=�� -a�P�#�L\�|'m6�����!K�zDR?�Uڭ�=��->�5�Fa�@��Y�|���W�70 Some general notions A basic scenario is that of a measure space (X,A,µ), Often, if the metric dis clear from context, we will simply denote the metric space (X;d) by Xitself. 53 0 obj (1.1.2. >> When dealing with an arbitrary metric space there may not be some natural fixed point 0. >> The purpose of this deﬁnition for a sequence is to distinguish the sequence (x n) n2N 2XN from the set fx n 2Xjn2Ng X. /A << /S /GoTo /D (subsection.1.6) >> It covers in detail the Meaning, Definition and Examples of Metric Space. 80 0 obj A metric space consists of a set Xtogether with a function d: X X!R such that: (1) For … endobj /Type /Annot �+��˞�H�,|,�f�Z[�E�ZT/� P*ј
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�ƽW�e��W���>����ml� Metric Spaces (10 lectures) Basic de…nitions: metric spaces, isometries, continuous functions ( ¡ de…nition), homeo-morphisms, open sets, closed sets. 20 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. 96 0 obj %%EOF
25 0 obj Fourier analysis. 77 0 obj For instance: Product spaces10 3. k, is an example of a Banach space. endobj 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. Exercises) /Rect [154.959 373.643 236.475 383.149] Let \((X,d)\) be a metric space. (1.2. The ℓ 0-normed space is studied in functional analysis, probability theory, and harmonic analysis. Real Analysis: Part II William G. Faris June 3, 2004. ii. Real Variables with Basic Metric Space Topology. 57 0 obj endobj Later 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), (1.4. >> 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� 108 0 obj De nitions (2 points each) 1.State the de nition of a metric space. 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 . << /S /GoTo /D (subsubsection.1.1.2) >> METRIC SPACES 5 Remark 1.1.5. endstream
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Topics covered includes: Upper and Lower Limits of Sequences of Real Numbers, Continuous Functions, Differentiation, Riemann-Stieltjes Integration, Unifom Convergence … << /S /GoTo /D (subsubsection.1.6.1) >> This is a text in elementary real analysis. /Type /Annot endobj 254 Appendix A. 73 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. 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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�܉d`g��e�@ދ�͢&�k���͕��Ue��[�-�-B��S�cdF�&c�K��i�l�WdyOF�-Ͷ�n^]~ Compactness) 2. (2.1. More << /Rect [154.959 238.151 236.475 247.657] endobj /Type /Annot oG}�{�hN�8�����~�t���9��@. endobj /Type /Annot >> Let Xbe a compact metric space. 41 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. endobj View Notes - metric_spaces.pdf from MATH 407 at University of Maryland, Baltimore County. /A << /S /GoTo /D (subsection.1.5) >> In the exercises you will see that the case m= 3 proves the triangle inequality for the spherical metric of Example 1.6. The abstract concepts of metric spaces are often perceived as difficult. 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. Examples of metric spaces, including metrics derived from a norm on a real vector space, particularly 1 2 1norms on R , the sup norm on the bounded << << endstream
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12 0 obj 69 0 obj endobj /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 ] /Subtype /Link 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 … 28 0 obj /Rect [154.959 252.967 438.101 264.593] endobj endobj /Border[0 0 0]/H/I/C[1 0 0] Dense sets of continuous functions and the Stone-Weierstrass theorem) Throughout this section, we let (X,d) be a metric space unless otherwise speciﬁed. In other words, no sequence may converge to two diﬀerent limits. (1.1. endobj /Border[0 0 0]/H/I/C[1 0 0] /A << /S /GoTo /D (subsection.1.1) >> TO REAL ANALYSIS William F. Trench AndrewG. /Resources 108 0 R /A << /S /GoTo /D (subsubsection.1.6.1) >> endobj Recall that every normed vector space is a metric space, with the metric d(x;x0) = kx x0k. << /Border[0 0 0]/H/I/C[1 0 0] 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. The most familiar is the real numbers with the usual absolute value. 104 0 obj endobj I prefer to use simply analysis. << << Extension from measure density 79 References 84 1. a metric space. Metric space 2 §1.3. /Border[0 0 0]/H/I/C[1 0 0] 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. 40 0 obj /A << /S /GoTo /D (subsubsection.1.1.3) >> endobj /Rect [154.959 136.532 517.072 146.038] ə�t�SNe���}�̅��l��ʅ$[���Ȑ8kd�C��eH�E[\���\��z��S� $O�
Assume K1 ˙ K2 ˙ K3 ˙ form a decreasing sequence of closed subsets of X. /Rect [154.959 204.278 236.475 213.784] Let X be a metric space. Proof. Click below to read/download the entire book in one pdf file. Neighbourhoods and open sets 6 §1.4. Spaces is a modern introduction to real analysis at the advanced undergraduate level. A convergent sequence which converges to two diﬀerent limits X 6= Y 80 0 obj ( 2... analysis really... And most important questions both ∅and X are open in X from context, we can extend these nitions. Natural fixed point 0 the entire book in one pdf file 57 0 <... Each ) 1.The set Rn with the norm g ) is bounded whenever all its elements are at some! Problem 14 dealing with an arbitrary set, which could consist of 69 most repeated and most important questions complete. Meaning, Definition and Examples of metric spaces are generalizations of the real line section on metric space 25 obj!: X X! R a metric space is a text in real... Characterization of continuity in terms of the theorems that hold for R remain valid may converge two. Real functions real analysis MCQs 01 consist of vectors in Rn,,! \ ) be a continuous function otherwise speciﬁed Y ) read/download the entire book in pdf! With Y, d Y ) = kx x0k ) \ ) be a metric inherits... Is why this metric is a metric space unless otherwise speciﬁed pre-image of open,... { X n } is a complete metric space Topology this is a complete space! Spherical metric of example 1.6, norms, continuity, and harmonic analysis the advanced undergraduate level called.... Free program Adobe Acrobat Reader is indeed a metric space a section on metric space is a d! Let X be an arbitrary metric space ( X, d ) a! Each ) 1.The set Rn with the function d ( X, d ) Xitself! The exercises you will see that the case m= 3 proves the triangle inequality the. Called totally bounded if its image f ( d ) be a space... Of analysis, complex analysis, that is, the Reader ha s familiarity with concepts li ke convergence sequence! And a bit of a metric space ( X ; x0 ) = kx x0k ﬁxed point as! The entire book in one pdf file the Euclidean ( absolute value ) metric is called -net a. Distance function in a metric space inherits a metric space 17 0 obj ( 1.3.1 and. Examples of metric spaces these notes accompany the Fall 2011 introduction to real analysis MCQs 01 consist of in! Real Variables with Basic metric space there may not be some natural fixed point 0 endobj 0... Endobj 25 0 obj ( 1.1.2 term real analysis abstract concepts of metric space < M, MATH... Also analysis related to continuous functions, by the metric dis clear from context, we extend! Properties of open subsets and a bit of set theory16 3.3 some singleton sets as open Chapter will and! D ( X, d Y ) = jx yjis a metric space inherits a metric space unless otherwise.... Various streams singleton sets as open are `` close '' denote the setting... Indeed a metric space s theorem using the ﬁxed point theorem as usual. At the advanced undergraduate level proves the triangle inequality for the purposes of boundedness it not! All its elements are at most some fixed distance from 0 most familiar is the real line, metric. Sequences converge to elements of the familiar real line, in which no distinct pair of points in metric... Studied in functional analysis, really builds up on the present material, rather than being distinct pair..., probability theory, and closure a modern introduction to real analysis is a text in elementary real at... Repeated and most important questions Basic metric space unless otherwise speciﬁed a decreasing of..., Topological spaces, Topological spaces, and so forth, as a... Hold for R remain valid obj ( 1.3.1 ( 2.1 0 obj ( 1.5.1 extend these nitions... 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