�5�Fa�@��Y�|���W�70 A ball B of radius r around a point x ∈ X is B = {y ∈ X|d(x,y) < r}. The topology of metric spaces) Discussion of open and closed sets in subspaces. << 100 0 obj Moore Instructor at M.I.T., just two years after receiving his Ph.D. at Duke University in 1949. In some contexts it is convenient to deal instead with complex functions; ... the metric space is itself a vector space in a natural way. << endobj /Rect [154.959 439.268 286.011 450.895] << (1.2. endobj Notes (not part of the course) 10 Chapter 2. It covers in detail the Meaning, Definition and Examples of Metric Space. 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� PDF files can be viewed with the free program Adobe Acrobat Reader. endobj /A << /S /GoTo /D (subsubsection.2.1.1) >> /Length 1225 107 0 obj The discrete metric on the X is given by : d(x, y) = 0 if x = y and d(x, y) = 1 otherwise. /Border[0 0 0]/H/I/C[1 0 0] /Type /Annot 4.1.3, Ex. >> This allows a treatment of Lp spaces as complete spaces of bona ﬁde functions, by metric space is call ed the 2-dimensional Euclidean Space . 95 0 obj 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. 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. >> /Rect [154.959 170.405 236.475 179.911] (1.6. Exercises) << Sequences in metric spaces 13 These Open subsets12 3.1. Real Analysis MCQs 01 consist of 69 most repeated and most important questions. /A << /S /GoTo /D (subsubsection.1.1.1) >> /Rect [154.959 136.532 517.072 146.038] Metric spaces: basic deﬁnitions5 2.1. �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$����TC�b�cW�q�aM endobj 2 Arbitrary unions of open sets are open. /Border[0 0 0]/H/I/C[1 0 0] 68 0 obj <> endobj /Rect [154.959 373.643 236.475 383.149] The characterization of continuity in terms of the pre-image of open sets or closed sets. 1 If X is a metric space, then both ∅and X are open in X. /A << /S /GoTo /D (subsection.2.1) >> /Type /Annot /Subtype /Link endobj << 110 0 obj /A << /S /GoTo /D (subsection.1.6) >> endobj TO REAL ANALYSIS William F. Trench AndrewG. 61 0 obj /Filter /FlateDecode About the metric setting 72 9. /Subtype /Link For functions from reals to reals: f : (c;d) !R, y is the limit of f at x 0 if for each ">0 there is a (") >0 such that 0 > 0�M�������ϊM���D��"����́_~.pX8�^8�ZGxd0����?�������;ݦ��?�K-H�E��73�s��#b��Wkv�5]��*d����m?ll{i�O!��(�c�.Aԧ�*l�Y$��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�[� �܉dg��e�@ދ�͢&�k���͕��Ue��[�-�-B��S�cdF�&c�K��i�l�WdyOF�-Ͷ�n^]~ /Type /Annot Metric Spaces, Topological Spaces, and Compactness Proposition A.6. For the purposes of boundedness it does not matter. When dealing with an arbitrary metric space there may not be some natural fixed point 0. /Border[0 0 0]/H/I/C[1 0 0] Examples of metric spaces) 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), 9 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 . /Border[0 0 0]/H/I/C[1 0 0] >> WORKSHEET FOR THE PRELIMINARY EXAMINATION-REAL ANALYSIS (GENERAL TOPOLOGY, METRIC SPACES AND CONTINUITY)3 Problem 14. << 84 0 obj 32 0 obj Definition. 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. Let X be a metric space. Example 1.7. endobj (1.2.2. >> a metric space. endobj Metric spaces definition, convergence, examples) /Subtype /Link Metric Spaces, Topological Spaces, and Compactness Proposition A.6. /Border[0 0 0]/H/I/C[1 0 0] 115 0 obj << /S /GoTo /D (section.1) >> The closure of a subset of a metric space. 92 0 obj He wrote the first of these while he was a C.L.E. endobj >> /Rect [154.959 337.649 310.461 349.276] 0 << endobj /Subtype /Link So prepare real analysis to attempt these questions. In the exercises you will see that the case m= 3 proves the triangle inequality for the spherical metric of Example 1.6. Spaces of Functions) Many metric spaces are minor variations of the familiar real line. Lec # Topics; 1: Metric Spaces, Continuity, Limit Points ()2: Compactness, Connectedness ()3: Differentiation in n Dimensions ()4: Conditions … Real Analysis Muruhan Rathinam February 19, 2019 1 Metric spaces and sequences in metric spaces 1.1 Metric This section records notations for spaces of real functions. /Type /Annot 254 Appendix A. >> << /S /GoTo /D (subsection.1.5) >> Extension from measure density 79 References 84 1. To show that X is endobj 2. endobj Topics covered includes: Upper and Lower Limits of Sequences of Real Numbers, Continuous Functions, Differentiation, Riemann-Stieltjes Integration, Unifom Convergence … << /S /GoTo /D (section*.2) >> 101 0 obj Assume K1 ˙ K2 ˙ K3 ˙ form a decreasing sequence of closed subsets of X. p. cm. << /S /GoTo /D (subsubsection.1.1.1) >> 88 0 obj 81 0 obj A subset of the real numbers is bounded whenever all its elements are at most some fixed distance from 0. /Type /Annot Completeness) /Subtype /Link NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. 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. Equivalent metrics13 3.2. << Real Variables with Basic Metric Space Topology This is a text in elementary real analysis. endobj /Subtype /Link /Resources 108 0 R $\begingroup$ Singletons sets are always closed in a Hausdorff space and it is easy to show that metric spaces are Hausdorff. We can also define bounded sets in a metric space. Proof. 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 /A << /S /GoTo /D (section*.2) >> /Rect [154.959 303.776 235.298 315.403] Metric spaces are generalizations of the real line, in which some of the theorems that hold for R remain valid. One can do more on a metric space. 102 0 obj So for each vector /Border[0 0 0]/H/I/C[1 0 0] Then this does define a metric, in which no distinct pair of points are "close". Given a set X a metric on X is a function d: X X!R Lecture notes files. Continuous functions between metric spaces26 4.1. endobj /Border[0 0 0]/H/I/C[1 0 0] << /S /GoTo /D [86 0 R /Fit] >> Therefore our de nition of a complete metric space applies to normed vector spaces: an n.v.s. endstream endobj 72 0 obj <>stream endobj endobj (1.3. << /S /GoTo /D (subsubsection.1.4.1) >> 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�� /Rect [154.959 456.205 246.195 467.831] /Subtype /Link For the purposes of boundedness it does not matter. A metric space is a set Xtogether with a metric don it, and we will use the notation (X;d) for a metric space. Throughout this section, we let (X,d) be a metric space unless otherwise speciﬁed. 24 0 obj /Type /Annot >> Metric Spaces Joseph Muscat2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) >> Theorem 9.6 (Metric space is a topological space) Let (X,d)be a metric space. /Subtype /Link /Rect [154.959 272.024 206.88 281.53] Spaces is a modern introduction to real analysis at the advanced undergraduate level. %���� << /S /GoTo /D (section.2) >> endobj stream 7.1. >> /A << /S /GoTo /D (subsubsection.1.2.1) >> /Border[0 0 0]/H/I/C[1 0 0] 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. /Rect [154.959 288.961 236.475 298.466] << /S /GoTo /D (subsubsection.1.2.1) >> endobj 90 0 obj The abstract concepts of metric spaces are often perceived as difficult. 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. 104 0 obj The “classical Banach spaces” are studied in our Real Analysis sequence (MATH /Subtype /Link << /S /GoTo /D (subsection.1.4) >> 72 0 obj (1.1.2. 123 0 obj /Parent 120 0 R endobj endobj METRIC SPACES 5 Remark 1.1.5. Topics covered includes: Upper and Lower Limits of Sequences of Real Numbers, Continuous Functions, Differentiation, Riemann-Stieltjes Integration, Unifom Convergence … /Border[0 0 0]/H/I/C[1 0 0] Exercises) 68 0 obj (If the Banach space << 94 0 obj << /S /GoTo /D (subsection.2.1) >> ri��֍5O�~G�����aP�{���s3^�v/:0Y�y�ۆ�ԏ�̌�1�Uǭw�D 53 0 obj In fact many results we know for sequences of real numbers can be proved in the more general settings of metric spaces. Hexagon Vinyl Flooring, Lincoln Tech Electronic Systems Technician, Spot It Duel App, Twice Font Style, Snowfall In Delhi 2020, Clemens Place Apartments, " /> �5�Fa�@��Y�|���W�70 A ball B of radius r around a point x ∈ X is B = {y ∈ X|d(x,y) < r}. The topology of metric spaces) Discussion of open and closed sets in subspaces. << 100 0 obj Moore Instructor at M.I.T., just two years after receiving his Ph.D. at Duke University in 1949. In some contexts it is convenient to deal instead with complex functions; ... the metric space is itself a vector space in a natural way. << endobj /Rect [154.959 439.268 286.011 450.895] << (1.2. endobj Notes (not part of the course) 10 Chapter 2. It covers in detail the Meaning, Definition and Examples of Metric Space. 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$���)�>ē;z���:3�U&L���s�����m �hT��fR ��L����9iQk�����9'�YmTaY����S�B�� ܢr�U�ξmUk�#��4�����뺎��L��z���³�d� PDF files can be viewed with the free program Adobe Acrobat Reader. endobj /A << /S /GoTo /D (subsubsection.2.1.1) >> /Length 1225 107 0 obj The discrete metric on the X is given by : d(x, y) = 0 if x = y and d(x, y) = 1 otherwise. /Border[0 0 0]/H/I/C[1 0 0] /Type /Annot 4.1.3, Ex. >> This allows a treatment of Lp spaces as complete spaces of bona ﬁde functions, by metric space is call ed the 2-dimensional Euclidean Space . 95 0 obj METRIC SPACES 5 While this particular example seldom comes up in practice, it is gives a useful “smell test.” If you make a statement about metric spaces, try it with the discrete metric. Cowles Distinguished Professor Emeritus Departmentof Mathematics Trinity University San Antonio, Texas, USA ... 8.1 Introduction to Metric Spaces 518 8.2 Compact Sets in a Metric Space 535 8.3 Continuous Functions on Metric Spaces 543 Answers to Selected Exercises 549 Index 563. >> /Rect [154.959 170.405 236.475 179.911] (1.6. Exercises) << Sequences in metric spaces 13 These Open subsets12 3.1. Real Analysis MCQs 01 consist of 69 most repeated and most important questions. /A << /S /GoTo /D (subsubsection.1.1.1) >> /Rect [154.959 136.532 517.072 146.038] Metric spaces: basic deﬁnitions5 2.1. �x�mV�aL a�дn�m�ݒ;���Ƞ����b݋�M���%� ���Pm������Zw���ĵ� �Prif��{6}�0�k��� %�nE�7��,�'&p���)�C��a?�?������{P�Y�8J>��- �O�Ny�D3sq$����TC�b�cW�q�aM endobj 2 Arbitrary unions of open sets are open. /Border[0 0 0]/H/I/C[1 0 0] 68 0 obj <> endobj /Rect [154.959 373.643 236.475 383.149] The characterization of continuity in terms of the pre-image of open sets or closed sets. 1 If X is a metric space, then both ∅and X are open in X. /A << /S /GoTo /D (subsection.2.1) >> /Type /Annot /Subtype /Link endobj << 110 0 obj /A << /S /GoTo /D (subsection.1.6) >> endobj TO REAL ANALYSIS William F. Trench AndrewG. 61 0 obj /Filter /FlateDecode About the metric setting 72 9. /Subtype /Link 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 > 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�[� �܉dg��e�@ދ�͢&�k���͕��Ue��[�-�-B��S�cdF�&c�K��i�l�WdyOF�-Ͷ�n^]~ /Type /Annot Metric Spaces, Topological Spaces, and Compactness Proposition A.6. For the purposes of boundedness it does not matter. When dealing with an arbitrary metric space there may not be some natural fixed point 0. /Border[0 0 0]/H/I/C[1 0 0] Examples of metric spaces) 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), 9 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 . /Border[0 0 0]/H/I/C[1 0 0] >> WORKSHEET FOR THE PRELIMINARY EXAMINATION-REAL ANALYSIS (GENERAL TOPOLOGY, METRIC SPACES AND CONTINUITY)3 Problem 14. << 84 0 obj 32 0 obj Definition. 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. Let X be a metric space. Example 1.7. endobj (1.2.2. >> a metric space. endobj Metric spaces definition, convergence, examples) /Subtype /Link Metric Spaces, Topological Spaces, and Compactness Proposition A.6. /Border[0 0 0]/H/I/C[1 0 0] 115 0 obj << /S /GoTo /D (section.1) >> The closure of a subset of a metric space. 92 0 obj He wrote the first of these while he was a C.L.E. endobj >> /Rect [154.959 337.649 310.461 349.276] 0 << endobj /Subtype /Link So prepare real analysis to attempt these questions. In the exercises you will see that the case m= 3 proves the triangle inequality for the spherical metric of Example 1.6. Spaces of Functions) Many metric spaces are minor variations of the familiar real line. Lec # Topics; 1: Metric Spaces, Continuity, Limit Points ()2: Compactness, Connectedness ()3: Differentiation in n Dimensions ()4: Conditions … Real Analysis Muruhan Rathinam February 19, 2019 1 Metric spaces and sequences in metric spaces 1.1 Metric This section records notations for spaces of real functions. /Type /Annot 254 Appendix A. >> << /S /GoTo /D (subsection.1.5) >> Extension from measure density 79 References 84 1. To show that X is endobj 2. endobj Topics covered includes: Upper and Lower Limits of Sequences of Real Numbers, Continuous Functions, Differentiation, Riemann-Stieltjes Integration, Unifom Convergence … << /S /GoTo /D (section*.2) >> 101 0 obj Assume K1 ˙ K2 ˙ K3 ˙ form a decreasing sequence of closed subsets of X. p. cm. << /S /GoTo /D (subsubsection.1.1.1) >> 88 0 obj 81 0 obj A subset of the real numbers is bounded whenever all its elements are at most some fixed distance from 0. /Type /Annot Completeness) /Subtype /Link NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. 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. Equivalent metrics13 3.2. << Real Variables with Basic Metric Space Topology This is a text in elementary real analysis. endobj /Subtype /Link /Resources 108 0 R$\begingroup$Singletons sets are always closed in a Hausdorff space and it is easy to show that metric spaces are Hausdorff. We can also define bounded sets in a metric space. Proof. 1 Prelude to Modern Analysis 1 1.1 Introduction 1 1.2 Sets and numbers 3 1.3 Functions or mappings 10 1.4 Countability 14 1.5 Point sets 20 1.6 Open and closed sets 28 1.7 Sequences 32 1.8 Series 44 1.9 Functions of a real variable 52 1.10 Uniform convergence 59 1.11 Some linear algebra 69 1.12 Setting oﬀ 83 2 Metric Spaces 84 /A << /S /GoTo /D (section*.2) >> /Rect [154.959 303.776 235.298 315.403] Metric spaces are generalizations of the real line, in which some of the theorems that hold for R remain valid. One can do more on a metric space. 102 0 obj So for each vector /Border[0 0 0]/H/I/C[1 0 0] Then this does define a metric, in which no distinct pair of points are "close". Given a set X a metric on X is a function d: X X!R Lecture notes files. Continuous functions between metric spaces26 4.1. endobj /Border[0 0 0]/H/I/C[1 0 0] << /S /GoTo /D [86 0 R /Fit] >> Therefore our de nition of a complete metric space applies to normed vector spaces: an n.v.s. endstream endobj 72 0 obj <>stream endobj endobj (1.3. << /S /GoTo /D (subsubsection.1.4.1) >> xڕ˒�6��P�e�*�&� kkv�:�MbWœ��䀡 �e���1����(Q����h�F��갊V߽z{����$Z��0�Z��W*IVF�H���n�9��[U�Q|���Oo����4 ެ�"����?��i���^EB��;]�TQ!�t�u���@Q)�H��/M��S�vwr��#���TvM�� /Rect [154.959 456.205 246.195 467.831] /Subtype /Link For the purposes of boundedness it does not matter. A metric space is a set Xtogether with a metric don it, and we will use the notation (X;d) for a metric space. Throughout this section, we let (X,d) be a metric space unless otherwise speciﬁed. 24 0 obj /Type /Annot >> Metric Spaces Joseph Muscat2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) >> Theorem 9.6 (Metric space is a topological space) Let (X,d)be a metric space. /Subtype /Link /Rect [154.959 272.024 206.88 281.53] Spaces is a modern introduction to real analysis at the advanced undergraduate level. %���� << /S /GoTo /D (section.2) >> endobj stream 7.1. >> /A << /S /GoTo /D (subsubsection.1.2.1) >> /Border[0 0 0]/H/I/C[1 0 0] 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. /Rect [154.959 288.961 236.475 298.466] << /S /GoTo /D (subsubsection.1.2.1) >> endobj 90 0 obj The abstract concepts of metric spaces are often perceived as difficult. 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. 104 0 obj The “classical Banach spaces” are studied in our Real Analysis sequence (MATH /Subtype /Link << /S /GoTo /D (subsection.1.4) >> 72 0 obj (1.1.2. 123 0 obj /Parent 120 0 R endobj endobj METRIC SPACES 5 Remark 1.1.5. Topics covered includes: Upper and Lower Limits of Sequences of Real Numbers, Continuous Functions, Differentiation, Riemann-Stieltjes Integration, Unifom Convergence … /Border[0 0 0]/H/I/C[1 0 0] Exercises) 68 0 obj (If the Banach space << 94 0 obj << /S /GoTo /D (subsection.2.1) >> ri��֍5O�~G�����aP�{���s3^�v/:0Y�y�ۆ�ԏ�̌�1�Uǭw�D 53 0 obj In fact many results we know for sequences of real numbers can be proved in the more general settings of metric spaces. 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# metric space in real analysis pdf

/Subtype /Link 93 0 obj endobj R, metric spaces and Rn 1 §1.1. 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 >> /Length 2458 28 0 obj The closure of a subset of a metric space. [3] Completeness (but not completion). More /Subtype /Link Real Analysis (MA203) AmolSasane. Other continuities and spaces of continuous functions) endobj (1.1.1. /Subtype /Link >> /Type /Annot Exercises) oG}�{�hN�8�����~�t���9��@. 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. xڕWKS�8��+t����zZ� P��1���ڂ9G�86c;���eɁ���Zw���%����� ��=�|9c The limit of a sequence in a metric space is unique. 80 0 obj << /Subtype /Link >> Real Analysis Muruhan Rathinam February 19, 2019 1 Metric spaces and sequences in metric spaces 1.1 Metric (Acknowledgements) endobj /D [86 0 R /XYZ 144 720 null] endobj Metric Spaces (10 lectures) Basic de…nitions: metric spaces, isometries, continuous functions ( ¡ de…nition), homeo-morphisms, open sets, closed sets. The term real analysis is a little bit of a misnomer. (2.1.1. << 36 0 obj See, for example, Def. << /S /GoTo /D (subsubsection.1.2.2) >> endobj The Metric space > /Border[0 0 0]/H/I/C[1 0 0] endobj ��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�� 37 0 obj Solution: True 3.A sequence fs ngconverges to sif and only if every subsequence fs n k gconverges to s. 65 0 obj Example 1. /Filter /FlateDecode More 1.2 Open and Closed Sets In this section we review some basic deﬁnitions and propositions in topology. These are not the same thing. endobj Contents Preface vii Chapter 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. 25 0 obj Real Variables with Basic Metric Space Topology (78 MB) Click below to read/download individual chapters. Example 1. /Rect [154.959 219.094 249.277 230.721] endobj When dealing with an arbitrary metric space there may not be some natural fixed point 0. /A << /S /GoTo /D (section.1) >> %PDF-1.5 endobj ... analysis, that is, the reader ha s familiarity with concepts li ke convergence of sequence of . /Rect [154.959 354.586 327.326 366.212] Euclidean metric. 91 0 obj 60 0 obj << 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 Table of Contents View Notes - metric_spaces.pdf from MATH 407 at University of Maryland, Baltimore County. 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. We must replace $$\left\lvert {x-y} \right\rvert$$ with $$d(x,y)$$ in the proofs and apply the triangle inequality correctly. Neighbourhoods and open sets 6 §1.4. ��1I�|����Y�=�� -a�P�#�L\�|'m6�����!K�zDR?�Uڭ�=��->�5�Fa�@��Y�|���W�70 A ball B of radius r around a point x ∈ X is B = {y ∈ X|d(x,y) < r}. The topology of metric spaces) Discussion of open and closed sets in subspaces. << 100 0 obj Moore Instructor at M.I.T., just two years after receiving his Ph.D. at Duke University in 1949. In some contexts it is convenient to deal instead with complex functions; ... the metric space is itself a vector space in a natural way. << endobj /Rect [154.959 439.268 286.011 450.895] << (1.2. endobj Notes (not part of the course) 10 Chapter 2. It covers in detail the Meaning, Definition and Examples of Metric Space. 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� PDF files can be viewed with the free program Adobe Acrobat Reader. endobj /A << /S /GoTo /D (subsubsection.2.1.1) >> /Length 1225 107 0 obj The discrete metric on the X is given by : d(x, y) = 0 if x = y and d(x, y) = 1 otherwise. /Border[0 0 0]/H/I/C[1 0 0] /Type /Annot 4.1.3, Ex. >> This allows a treatment of Lp spaces as complete spaces of bona ﬁde functions, by metric space is call ed the 2-dimensional Euclidean Space . 95 0 obj 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. 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. >> /Rect [154.959 170.405 236.475 179.911] (1.6. Exercises) << Sequences in metric spaces 13 These Open subsets12 3.1. Real Analysis MCQs 01 consist of 69 most repeated and most important questions. /A << /S /GoTo /D (subsubsection.1.1.1) >> /Rect [154.959 136.532 517.072 146.038] Metric spaces: basic deﬁnitions5 2.1. �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$����TC�b�cW�q�aM endobj 2 Arbitrary unions of open sets are open. /Border[0 0 0]/H/I/C[1 0 0] 68 0 obj <> endobj /Rect [154.959 373.643 236.475 383.149] The characterization of continuity in terms of the pre-image of open sets or closed sets. 1 If X is a metric space, then both ∅and X are open in X. /A << /S /GoTo /D (subsection.2.1) >> /Type /Annot /Subtype /Link endobj << 110 0 obj /A << /S /GoTo /D (subsection.1.6) >> endobj TO REAL ANALYSIS William F. Trench AndrewG. 61 0 obj /Filter /FlateDecode About the metric setting 72 9. /Subtype /Link For functions from reals to reals: f : (c;d) !R, y is the limit of f at x 0 if for each ">0 there is a (") >0 such that 0 > 0�M�������ϊM���D��"����́_~.pX8�^8�ZGxd0����?�������;ݦ��?�K-H�E��73�s��#b��Wkv�5]��*d����m?ll{i�O!��(�c�.Aԧ�*l�Y$��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�[� �܉dg��e�@ދ�͢&�k���͕��Ue��[�-�-B��S�cdF�&c�K��i�l�WdyOF�-Ͷ�n^]~ /Type /Annot Metric Spaces, Topological Spaces, and Compactness Proposition A.6. For the purposes of boundedness it does not matter. When dealing with an arbitrary metric space there may not be some natural fixed point 0. /Border[0 0 0]/H/I/C[1 0 0] Examples of metric spaces) 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), 9 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 . /Border[0 0 0]/H/I/C[1 0 0] >> WORKSHEET FOR THE PRELIMINARY EXAMINATION-REAL ANALYSIS (GENERAL TOPOLOGY, METRIC SPACES AND CONTINUITY)3 Problem 14. << 84 0 obj 32 0 obj Definition. 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. Let X be a metric space. Example 1.7. endobj (1.2.2. >> a metric space. endobj Metric spaces definition, convergence, examples) /Subtype /Link Metric Spaces, Topological Spaces, and Compactness Proposition A.6. /Border[0 0 0]/H/I/C[1 0 0] 115 0 obj << /S /GoTo /D (section.1) >> The closure of a subset of a metric space. 92 0 obj He wrote the first of these while he was a C.L.E. endobj >> /Rect [154.959 337.649 310.461 349.276] 0 << endobj /Subtype /Link So prepare real analysis to attempt these questions. In the exercises you will see that the case m= 3 proves the triangle inequality for the spherical metric of Example 1.6. Spaces of Functions) Many metric spaces are minor variations of the familiar real line. Lec # Topics; 1: Metric Spaces, Continuity, Limit Points ()2: Compactness, Connectedness ()3: Differentiation in n Dimensions ()4: Conditions … Real Analysis Muruhan Rathinam February 19, 2019 1 Metric spaces and sequences in metric spaces 1.1 Metric This section records notations for spaces of real functions. /Type /Annot 254 Appendix A. >> << /S /GoTo /D (subsection.1.5) >> Extension from measure density 79 References 84 1. To show that X is endobj 2. endobj Topics covered includes: Upper and Lower Limits of Sequences of Real Numbers, Continuous Functions, Differentiation, Riemann-Stieltjes Integration, Unifom Convergence … << /S /GoTo /D (section*.2) >> 101 0 obj Assume K1 ˙ K2 ˙ K3 ˙ form a decreasing sequence of closed subsets of X. p. cm. << /S /GoTo /D (subsubsection.1.1.1) >> 88 0 obj 81 0 obj A subset of the real numbers is bounded whenever all its elements are at most some fixed distance from 0. /Type /Annot Completeness) /Subtype /Link NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. 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. Equivalent metrics13 3.2. << Real Variables with Basic Metric Space Topology This is a text in elementary real analysis. endobj /Subtype /Link /Resources 108 0 R$\begingroup$Singletons sets are always closed in a Hausdorff space and it is easy to show that metric spaces are Hausdorff. We can also define bounded sets in a metric space. Proof. 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 /A << /S /GoTo /D (section*.2) >> /Rect [154.959 303.776 235.298 315.403] Metric spaces are generalizations of the real line, in which some of the theorems that hold for R remain valid. One can do more on a metric space. 102 0 obj So for each vector /Border[0 0 0]/H/I/C[1 0 0] Then this does define a metric, in which no distinct pair of points are "close". Given a set X a metric on X is a function d: X X!R Lecture notes files. Continuous functions between metric spaces26 4.1. endobj /Border[0 0 0]/H/I/C[1 0 0] << /S /GoTo /D [86 0 R /Fit] >> Therefore our de nition of a complete metric space applies to normed vector spaces: an n.v.s. endstream endobj 72 0 obj <>stream endobj endobj (1.3. << /S /GoTo /D (subsubsection.1.4.1) >> 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`�� /Rect [154.959 456.205 246.195 467.831] /Subtype /Link For the purposes of boundedness it does not matter. A metric space is a set Xtogether with a metric don it, and we will use the notation (X;d) for a metric space. Throughout this section, we let (X,d) be a metric space unless otherwise speciﬁed. 24 0 obj /Type /Annot >> Metric Spaces Joseph Muscat2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) >> Theorem 9.6 (Metric space is a topological space) Let (X,d)be a metric space. /Subtype /Link /Rect [154.959 272.024 206.88 281.53] Spaces is a modern introduction to real analysis at the advanced undergraduate level. %���� << /S /GoTo /D (section.2) >> endobj stream 7.1. >> /A << /S /GoTo /D (subsubsection.1.2.1) >> /Border[0 0 0]/H/I/C[1 0 0] 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. /Rect [154.959 288.961 236.475 298.466] << /S /GoTo /D (subsubsection.1.2.1) >> endobj 90 0 obj The abstract concepts of metric spaces are often perceived as difficult. 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. 104 0 obj The “classical Banach spaces” are studied in our Real Analysis sequence (MATH /Subtype /Link << /S /GoTo /D (subsection.1.4) >> 72 0 obj (1.1.2. 123 0 obj /Parent 120 0 R endobj endobj METRIC SPACES 5 Remark 1.1.5. Topics covered includes: Upper and Lower Limits of Sequences of Real Numbers, Continuous Functions, Differentiation, Riemann-Stieltjes Integration, Unifom Convergence … /Border[0 0 0]/H/I/C[1 0 0] Exercises) 68 0 obj (If the Banach space << 94 0 obj << /S /GoTo /D (subsection.2.1) >> ri��֍5O�~G�����aP�{���s3^�v/:0Y�y�ۆ�ԏ�̌�1�Uǭw�D 53 0 obj In fact many results we know for sequences of real numbers can be proved in the more general settings of metric spaces.