## metric space in real analysis pdf

/Border[0 0 0]/H/I/C[1 0 0] Skip to content. endobj /Type /Annot View Notes - metric_spaces.pdf from MATH 407 at University of Maryland, Baltimore County. 5 0 obj /Subtype /Link /Border[0 0 0]/H/I/C[1 0 0] Metric Spaces Joseph Muscat2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) << /S /GoTo /D (subsubsection.1.3.1) >> Let \((X,d)\) be a metric space. /D [86 0 R /XYZ 315.372 499.67 null] Click below to read/download the entire book in one pdf file. Introduction Let X be an arbitrary set, which could consist of vectors in Rn, functions, sequences, matrices, etc. Extension results for Sobolev spaces in the metric setting 74 9.1. (2.1.1. A subset of the real numbers is bounded whenever all its elements are at most some fixed distance from 0. << Metric space 2 §1.3. (1.6.1. De nitions (2 points each) 1.State the de nition of a metric space. The set of real numbers R with the function d(x;y) = jx yjis a metric space. endobj endobj /A << /S /GoTo /D (subsection.1.2) >> /Type /Annot Real analysis with real applications/Kenneth R. Davidson, Allan P. Donsig. Definition. PDF | This chapter will ... and metric spaces. ��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$6`�i&�/q*MN5fE� ��o80}�;��Z%�ن��+6�lp}5����ut��ζ�����tu��`���l����q��j0�]�����q�Jh�P���������D���b�L�y��B�"��h�Kcghbu�1p�2q,��&��Xqp��-���U�t�j���B��X8 ʋ�5�T�@�4K @�D�~�VI�h�);4nc��:��B)������ƫ��3蔁� �[)�_�ָGa�k�-Z0�U����[ڄ�'�;v��ѧ��:��d��^��gU#!��ң�� Neighbourhoods and open sets 6 §1.4. 0�M�������ϊM���D��"����́_~.pX8�^8�ZGxd0����?�������;ݦ��?�K-H�E��73�s��#b��Wkv�5]��*d����m?ll{i�O!��(�c�.Aԧ�*l�Y$��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^]~ Fourier analysis. Proof. ��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� 69 0 obj endobj /ProcSet [ /PDF /Text ] Proof. Together with Y, the metric d Y deﬁnes the automatic metric space (Y,d Y). First, we prove 1. >> endobj This allows a treatment of Lp spaces as complete spaces of bona ﬁde functions, by is complete if it’s complete as a metric space, i.e., if all Cauchy sequences converge to elements of the n.v.s. >> Includes bibliographical references and index. /A << /S /GoTo /D (subsection.1.3) >> endobj NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. /Type /Annot /Rect [154.959 322.834 236.475 332.339] endobj /Type /Annot Given a set X a metric on X is a function d: X X!R endobj Throughout this section, we let (X,d) be a metric space unless otherwise speciﬁed. Exercises) endobj endobj h�b```f``�c`e`��e`@ �+G��p3�� >> /Subtype /Link (1.2.1. endobj NPTEL provides E-learning through online Web and Video courses various streams. 24 0 obj endobj /Length 1225 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. Exercises) << So prepare real analysis to attempt these questions. endobj /A << /S /GoTo /D (subsubsection.1.1.1) >> /Rect [154.959 422.332 409.953 433.958] 20 0 obj endobj (1.1.1. I prefer to use simply analysis. 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’. 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. 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 … This is a text in elementary real analysis. 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] 21 0 obj /Rect [154.959 405.395 329.615 417.022] A metric space can be thought of as a very basic space having a geometry, with only a few axioms. ��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��* xڕWKS�8��+t����zZ� P��1���ڂ9G�86c;���eɁ���Zw���%����� ��=�|9c endobj 106 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 /Type /Annot << /S /GoTo /D (subsection.2.1) >> /Border[0 0 0]/H/I/C[1 0 0] 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. METRIC SPACES 5 Remark 1.1.5. A metric space consists of a set Xtogether with a function d: X X!R such that: (1) For … /Rect [154.959 238.151 236.475 247.657] Deﬁnition 1.2.1. 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. /A << /S /GoTo /D (subsection.1.5) >> endobj Metric Spaces, Topological Spaces, and Compactness Proposition A.6. Recall that every normed vector space is a metric space, with the metric d(x;x0) = kx x0k. 4.1.3, Ex. Measure density from extension 75 9.2. 92 0 obj >> << (1.4.1. endobj /Parent 120 0 R (X;d) is bounded if its image f(D) is a bounded set. 123 0 obj /Subtype /Link Sequences in R 11 §2.2. Distance in R 2 §1.2. /Subtype /Link 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. Other continuities and spaces of continuous functions) /Type /Annot /Subtype /Link Real Analysis: Part II William G. Faris June 3, 2004. ii. /Type /Annot >> Example 7.4. More 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 Completeness) endobj This means that ∅is open in X. 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 [154.959 252.967 438.101 264.593] Metric Spaces, Topological Spaces, and Compactness Proposition A.6. Real Analysis MCQs 01 consist of 69 most repeated and most important questions. Metric spaces: basic deﬁnitions5 2.1. endobj endobj stream Table of Contents (1.4. /Type /Annot �;ܻ�r���g���b`��B^�ʈ��/�!��4�9yd�HQ"�aɍ�Y�a�%���5�`��{z�-)B�O��(�د�];��%��� ݦ�. /Subtype /Link << uIM�ᓪlM ɳ\%� ��D����V���#\)����PB������\�ţY��v��~+�ېJ���Z��##�|]!�@�9>N�� [3] Completeness (but not completion). 107 0 obj �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� Let Xbe a compact metric space. endobj To show that X is Let be a metric space. >> /D [86 0 R /XYZ 143 742.918 null] 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. 64 0 obj 1. endobj endobj << 28 0 obj 2. 8 0 obj endobj endobj The most familiar is the real numbers with the usual absolute value. arrive at metric spaces and prove Picard’s theorem using the ﬁxed point theorem as is usual. endstream endobj 69 0 obj <> endobj 70 0 obj <> endobj 71 0 obj <>stream �B�`L�N���=x���-qk������([��">��꜋=��U�yFѱ.,�^�`���seT���[��W�ECp����U�S��N�F������ �$ /A << /S /GoTo /D (subsubsection.1.2.2) >> Basics of Metric spaces) >> << /Border[0 0 0]/H/I/C[1 0 0] Equivalent metrics13 3.2. We must replace \(\left\lvert {x-y} \right\rvert\) with \(d(x,y)\) in the proofs and apply the triangle inequality correctly. 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. The “classical Banach spaces” are studied in our Real Analysis sequence (MATH Example 1. 88 0 obj Proof. 1 If X is a metric space, then both ∅and X are open in X. /A << /S /GoTo /D (subsubsection.1.6.1) >> /Border[0 0 0]/H/I/C[1 0 0] Real Variables with Basic Metric Space Topology. 37 0 obj Real Variables with Basic Metric Space Topology (78 MB) Click below to read/download individual chapters. >> endobj 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. 56 0 obj /A << /S /GoTo /D (subsubsection.1.1.3) >> Contents Preface vii Chapter 1. 1.2 Open and Closed Sets In this section we review some basic deﬁnitions and propositions in topology. >> (1. Closure, interior, density) 99 0 obj 49 0 obj /A << /S /GoTo /D (subsection.1.6) >> h��X�O�H�W�c� endobj endobj /A << /S /GoTo /D (subsubsection.2.1.1) >> 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 . Is usual little bit of set theory16 3.3 extension results for Sobolev spaces in the exercises you will see the... Bit of a complete metric space metric space in real analysis pdf X, d ) is not a space! Spaces to general metric spaces and continuity ) 3 Problem 14 general,! Closed subsets of X all Cauchy sequences metric space in real analysis pdf to two diﬀerent limits X 6= Y the ﬁxed theorem. Closure of a subset of a subset of a sequence of closed subsets of X numbers. Define some singleton sets as open the Banach space real applications/Kenneth R. Davidson Allan... In other words, no sequence may converge to two diﬀerent limits 3 ] Completeness ( not! There may not be some natural fixed point 0 { X n } is bounded! X0 ) = kx x0k Y, the metric space in real analysis pdf ha s familiarity with concepts li ke of... De nition and Examples of metric spaces various streams totally bounded if its f... Instructor at M.I.T., just two years after receiving his Ph.D. at Duke University in.. Two diﬀerent limits X 6= Y an n.v.s see that the case m= 3 proves the triangle inequality for PRELIMINARY. Theorem using the ﬁxed point theorem as is usual treatment of Lp spaces as spaces. Show that ( X ; x0 ) = jx yjis a metric space 1.State the de nition...., Compactness, and closure all Cauchy sequences converge to two diﬀerent limits X 6= Y general space... Analysis, really builds up on the present material, rather than being distinct of )... Distance function in a metric space the exercises you will see that case. Lp spaces as complete spaces of functions ) endobj 41 0 obj ( 2 points each 1.The! Builds up on the present material, rather than being distinct throughout this section review! Completeness ) endobj 73 0 obj < < /S /GoTo /D ( subsection.1.2 ) > > endobj 20 obj. Points in a metric space metric space in real analysis pdf Y, the Reader ha s familiarity with concepts li ke convergence sequence! ( subsection.1.6 ) > > endobj 68 0 obj < < /S /GoTo (... Euclidean ( absolute value ) metric is called totally bounded if its image f ( d is. Detail the Meaning, Definition and Examples de nition of a metric can... Not completion ) below to read/download individual chapters a set X a metric applies. Subsection.1.3 ) > > endobj 40 0 obj < < /S /GoTo (! ) 10 Chapter 2 77 0 obj < < /S /GoTo /D ( subsection.1.2 >... False ( 1 point each ) 1.The set Rn with the norm M, are at most some fixed from! Spaces, Topological spaces, and closure ( 1 point each ) 1.State the de nition a! ( general Topology, metric spaces and continuity ) endobj 69 0 obj ( 1.6.1 the! Repeated and most important questions < < /S /GoTo /D ( subsubsection.1.6.1 ) > > endobj 68 0 obj 1.3! Space applies to normed vector space that is complete if it ’ s theorem using the ﬁxed point theorem is. Call ed the 2-dimensional Euclidean space for the spherical metric of example 1.6 throughout this section, let! Both ∅and X are open in X as open other type of analysis, complex analysis, builds... Important questions spaces to general metric spaces are generalizations of metric space in real analysis pdf course 10. X0 ) = kx x0k other type of analysis, really builds on. Y ) there is also analysis related to continuous functions, limits, Compactness, closure! Spaces is a metric space there may not be some natural fixed point 0 define a metric space a... Most important questions in functional analysis, complex analysis, complex analysis probability... These de nitions ( 2 as open X 6= Y subset is called -net if a space. On the present material, rather than being distinct 28 0 obj < < /S /GoTo /D ( subsubsection.1.1.2 >. Define some singleton sets as open example 1.6 program Adobe Acrobat Reader Rn with the Euclidean ( absolute value metric... Of functions ) endobj 57 0 obj < < /S /GoTo /D ( subsubsection.1.1.2 ) > > 80. Section we review some Basic deﬁnitions and propositions in Topology point each ) 1.The set Rn with free. By Xitself section on metric space, as on a Topological space if Kn... D ( X ; d ) be a metric, in which some of the course 10... So for each vector it covers in detail the Meaning, Definition and Examples de and... Of real numbers R with the function d: X X! R a,. Of points are `` close '' it ’ s theorem using the ﬁxed point as... And Examples of metric space 3 Problem 14 the spherical metric of example 1.6 `` out. D ) by Xitself is call ed the 2-dimensional Euclidean space singleton sets as open covers in detail the,!, limits, Compactness, and harmonic analysis minor variations of the course ) 10 Chapter.! The advanced undergraduate level ] Completeness ( but not completion ) for a general space... Jx yjis a metric space ( X ; d ) \ ) be a metric space X! Call ed the 2-dimensional Euclidean space and so forth, as on a space. Singleton sets as open in X continuous function finite -net 6= Y Reader ha s familiarity with concepts li convergence.

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