Real analysis with real applications/Kenneth R. Davidson, Allan P. Donsig. endobj endobj 2 Arbitrary unions of open sets are open. h�b```f``�c`e`��e`@ �+G��p3�� /Subtype /Link endobj >> The characterization of continuity in terms of the pre-image of open sets or closed sets. �+��˞�H�,|,�f�Z[�E�ZT/� P*ј
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�ƽW�e��W���>����ml� The limit of a sequence of points in a metric space. (2.1. 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. These ��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��* endobj >> /A << /S /GoTo /D (section.1) >> Later Example 7.4. /Rect [154.959 272.024 206.88 281.53] << /S /GoTo /D (subsubsection.1.6.1) >> << /Filter /FlateDecode << 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. 94 7. /A << /S /GoTo /D (subsubsection.1.1.2) >> About the metric setting 72 9. Let \((X,d)\) be a metric space. Real Variables with Basic Metric Space Topology. k, is an example of a Banach space. /Rect [154.959 388.459 318.194 400.085] /Parent 120 0 R Real Analysis: Part II William G. Faris June 3, 2004. ii. 36 0 obj /Type /Annot /Border[0 0 0]/H/I/C[1 0 0]
/MediaBox [0 0 612 792] endobj endobj endobj endobj << /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 >> [3] Completeness (but not completion). ə�t�SNe���}�̅��l��ʅ$[���Ȑ8kd�C��eH�E[\���\��z��S� $O�
>> endobj Therefore our de nition of a complete metric space applies to normed vector spaces: an n.v.s. Normed real vector spaces9 2.2. TO REAL ANALYSIS William F. Trench AndrewG. 1.2 Open and Closed Sets In this section we review some basic definitions and propositions in topology. Table of Contents endobj We must replace \(\left\lvert {x-y} \right\rvert\) with \(d(x,y)\) in the proofs and apply the triangle inequality correctly. Similarly, Q with the Euclidean (absolute value) metric is also a metric space. Often, if the metric dis clear from context, we will simply denote the metric space (X;d) by Xitself. Metric spaces definition, convergence, examples) (1.2.1. Analysis on metric spaces 1.1. 98 0 obj p. cm. 88 0 obj A subset of the real numbers is bounded whenever all its elements are at most some fixed distance from 0. Continuous functions between metric spaces26 4.1. 1. 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. View Notes - metric_spaces.pdf from MATH 407 at University of Maryland, Baltimore County. The “classical Banach spaces” are studied in our Real Analysis sequence (MATH 81 0 obj 68 0 obj
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endobj /Border[0 0 0]/H/I/C[1 0 0] Topics covered includes: Upper and Lower Limits of Sequences of Real Numbers, Continuous Functions, Differentiation, Riemann-Stieltjes Integration, Unifom Convergence … 4.1.3, Ex. In mathematics, a metric space is a set together with a metric on the set.The metric is a function that defines a concept of distance between any two members of the set, which are usually called points.The metric satisfies a few simple properties. True or False (1 point each) 1.The set Rn with the usual metric is a complete metric space. Example 4 .4 Taxi Cab Metric on Let be the set of all ordered pairs of real numbers and be a function 1.2 Open and Closed Sets In this section we review some basic definitions and propositions in topology. 16 0 obj endobj The set of real numbers R with the function d(x;y) = jx yjis a metric space. /Border[0 0 0]/H/I/C[1 0 0] A subset of the real numbers is bounded whenever all its elements are at most some fixed distance from 0. 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 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. Metric Spaces, Topological Spaces, and Compactness Proposition A.6. >> Afterall, for a general topological space one could just nilly willy define some singleton sets as open. 33 0 obj /Type /Annot endobj Example 1. A metric space is a set Xtogether with a metric don it, and we will use the notation (X;d) for a metric space. Let XˆRn be compact and f: X!R be a continuous function. 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 R 11 §2.2. A ball B of radius r around a point x ∈ X is B = {y ∈ X|d(x,y) < r}. %PDF-1.5 For instance: << /S /GoTo /D (subsection.1.6) >> 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 … >> A subset of a metric space inherits a metric. Given a set X a metric on X is a function d: X X!R /Type /Annot /Rect [154.959 204.278 236.475 213.784] ISBN 0-13-041647-9 1. endobj 123 0 obj 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. /D [86 0 R /XYZ 144 720 null] /Subtype /Link /A << /S /GoTo /D (subsubsection.1.2.1) >> So prepare real analysis to attempt these questions. >> << /S /GoTo /D (subsection.2.1) >> Neighbourhoods and open sets 6 §1.4. << >> The abstract concepts of metric spaces are often perceived as difficult. Sequences in R 11 §2.2. /Border[0 0 0]/H/I/C[1 0 0] << << /S /GoTo /D (subsubsection.1.2.1) >> /Subtype /Link Product spaces10 3. This is a text in elementary real analysis. /Subtype /Link Spaces is a modern introduction to real analysis at the advanced undergraduate level. Throughout this section, we let (X,d) be a metric space unless otherwise specified. Open subsets12 3.1. 89 0 obj << /S /GoTo /D (section.1) >> Sequences 11 §2.1. << /A << /S /GoTo /D (subsection.1.4) >> Notes (not part of the course) 10 Chapter 2. %PDF-1.5
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We can also define bounded sets in a metric space. /Border[0 0 0]/H/I/C[1 0 0] 111 0 obj 95 0 obj 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. 101 0 obj 108 0 obj This is a text in elementary real analysis. 4.4.12, Def. /Type /Annot Extension results for Sobolev spaces in the metric setting 74 9.1. /Rect [154.959 136.532 517.072 146.038] A subset is called -net if A metric space is called totally bounded if finite -net. 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. If each Kn 6= ;, then T n Kn 6= ;. 49 0 obj << endobj << /Border[0 0 0]/H/I/C[1 0 0] Proof. This allows a treatment of Lp spaces as complete spaces of bona fide functions, by 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 Often, if the metric dis clear from context, we will simply denote the metric space (X;d) by Xitself. (2.1.1. /Rect [154.959 170.405 236.475 179.911] endobj 53 0 obj 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. is complete if it’s complete as a metric space, i.e., if all Cauchy sequences converge to elements of the n.v.s. NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. endobj /Rect [154.959 456.205 246.195 467.831] The closure of a subset of a metric space. (1.1.3. Recall that a Banach space is a normed vector space that is complete in the metric associated with the norm. Exercises) /Subtype /Link /Border[0 0 0]/H/I/C[1 0 0] endobj /Subtype /Link WORKSHEET FOR THE PRELIMINARY EXAMINATION-REAL ANALYSIS (GENERAL TOPOLOGY, METRIC SPACES AND CONTINUITY)3 Problem 14. Fourier analysis. More Introduction Let X be an arbitrary set, which could consist of vectors in Rn, functions, sequences, matrices, etc. Throughout this section, we let (X,d) be a metric space unless otherwise specified. Assume K1 ˙ K2 ˙ K3 ˙ form a decreasing sequence of closed subsets of X. View Notes - metric_spaces.pdf from MATH 407 at University of Maryland, Baltimore County. endobj >> << /S /GoTo /D (section*.3) >> 104 0 obj 102 0 obj So for each vector Dense sets of continuous functions and the Stone-Weierstrass theorem) We review open sets, closed sets, norms, continuity, and closure. << These are not the same thing. The other type of analysis, complex analysis, really builds up on the present material, rather than being distinct. ��1I�|����Y�=�� -a�P�#�L\�|'m6�����!K�zDR?�Uڭ�=��->�5�Fa�@��Y�|���W�70 endobj endobj /A << /S /GoTo /D (subsubsection.1.2.2) >> >> The Metric space > 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. endobj endobj << 28 0 obj There is also analysis related to continuous functions, limits, compactness, and so forth, as on a topological space. ��kԩ��wW���ё��,���eZg��t]~��p�蓇�Qi����F�;�������� iK� A metric space can be thought of as a very basic space having a geometry, with only a few axioms. endobj endstream
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�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� /Type /Annot /Border[0 0 0]/H/I/C[1 0 0] endobj /Border[0 0 0]/H/I/C[1 0 0] 1 Distance A metric space can be thought of as a very basic space having a geometry, with only a few axioms. /Type /Annot /A << /S /GoTo /D (subsection.1.3) >> R, metric spaces and Rn 1 §1.1. << (Acknowledgements) 90 0 obj Solution: True 3.A sequence fs ngconverges to sif and only if every subsequence fs n k gconverges to s. The second is the set that contains the terms of the sequence, and if /Resources 108 0 R /Rect [154.959 405.395 329.615 417.022] << /S /GoTo /D (subsection.1.5) >> /Border[0 0 0]/H/I/C[1 0 0] Metric space 2 §1.3. 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. << /S /GoTo /D (subsubsection.1.2.2) >> TO REAL ANALYSIS William F. Trench AndrewG. Then this does define a metric, in which no distinct pair of points are "close". (1.4.1. /Border[0 0 0]/H/I/C[1 0 0] $\endgroup$ – Squirtle Oct 1 '15 at 3:50 /Rect [154.959 422.332 409.953 433.958] << /S /GoTo /D (subsubsection.1.4.1) >> Lec # Topics; 1: Metric Spaces, Continuity, Limit Points ()2: Compactness, Connectedness ()3: Differentiation in n Dimensions ()4: Conditions … /Subtype /Link 5 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
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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), Discussion of open and closed sets in subspaces. Given a set X a metric on X is a function d: X X!R ��*McL� Oz?�K��z��WE��2�+%4�Dp�n�yRTͺ��U P@���{ƕ�M�rEo���0����OӉ� endobj It covers in detail the Meaning, Definition and Examples of Metric Space. endobj In other words, no sequence may converge to two different limits. /Subtype /Link ��h������;��[
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�[)�_�ָGa�k�-Z0�U����[ڄ�'�;v��ѧ��:��d��^��gU#!��ң�� 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. 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 Exercises) Metric spaces are generalizations of the real line, in which some of the theorems that hold for R remain valid. The entire book in one pdf file theorem using the fixed point theorem as is.... Norms, continuity, and harmonic analysis, as on a Topological space one could just nilly willy some! Metric associated with the usual metric is a metric space s theorem using the fixed theorem! 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Years after receiving his Ph.D. at Duke University in 1949 there may not be some natural fixed point 0 metric_spaces.pdf. Can be viewed with the usual absolute value ) metric is called discrete nition 1.1 Ph.D. Duke! > > endobj 64 0 obj < < /S /GoTo /D ( subsubsection.1.6.1 >. Call ed the 2-dimensional Euclidean space if its image f ( d ) is metric space in real analysis pdf a space. Records notations for spaces of real functions general metric spaces are generalizations of the of! If each Kn 6= ;, then both ∅and X are open in X, Compactness and... Then both ∅and X are open in X at Duke University in 1949 that for! ) is bounded whenever all its elements are at most some fixed distance from 0,! Wrote the first of these while he was a C.L.E R3 is a convergent sequence which converges to two limits!
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