72 0 obj >> endobj 36 0 obj 254 Appendix A. xڕWKS�8��+t����zZ� P��1���ڂ9G�86c;���eɁ���Zw���%����� ��=�|9c Often, if the metric dis clear from context, we will simply denote the metric space (X;d) by Xitself. 87 0 obj Proof. (2.1. Real Variables with Basic Metric Space Topology. �+��˞�H�,|,�f�Z[�E�ZT/� P*ј � �ƽW�e��W���>����ml� (1.3.1. R, metric spaces and Rn 1 §1.1. Example 1. /Type /Annot endobj 2. 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. /D [86 0 R /XYZ 315.372 499.67 null] (1. Exercises) /Subtype /Link << /S /GoTo /D (subsubsection.1.1.2) >> Let \((X,d)\) be a metric space. Basics of Metric spaces) 5 0 obj << endobj We can also define bounded sets in a metric space. 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 [154.959 185.221 246.864 196.848] Lec # Topics; 1: Metric Spaces, Continuity, Limit Points ()2: Compactness, Connectedness ()3: Differentiation in n Dimensions ()4: Conditions … %PDF-1.5 Lecture notes files. The limit of a sequence of points in a metric space. �@� �YZ<5�e��SE� оs�~fx�u���� �Au�%���D]�,�Q�5�j�3���\�#�l��˖L�?�;8�5�@�{R[VS=���� A subset of the real numbers is bounded whenever all its elements are at most some fixed distance from 0. << endobj << endobj /Subtype /Link >> endobj So prepare real analysis to attempt these questions. 88 0 obj endstream /A << /S /GoTo /D (subsection.2.1) >> endobj endobj 123 0 obj /A << /S /GoTo /D (subsection.1.1) >> Example: Any bounded subset of 1. /Type /Annot 1 0 obj ��1I�|����Y�=�� -a�P�#�L\�|'m6�����!K�zDR?�Uڭ�=��->�5�Fa�@��Y�|���W�70 endobj ��kԩ��wW���ё��,���eZg��t]~��p�蓇�Qi����F�;�������� iK� Often, if the metric dis clear from context, we will simply denote the metric space (X;d) by Xitself. endobj (2.1.1. << << 109 0 obj endobj 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� /MediaBox [0 0 612 792] Let XˆRn be compact and f: X!R be a continuous function. The set of real numbers R with the function d(x;y) = jx yjis a metric space. 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. Real Variables with Basic Metric Space Topology This is a text in elementary real analysis. He wrote the first of these while he was a C.L.E. 17 0 obj /A << /S /GoTo /D (subsubsection.1.6.1) >> 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. 92 0 obj (1.4. METRIC SPACES 5 Remark 1.1.5. endobj Real Analysis (MA203) AmolSasane. 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. Metric Spaces, Topological Spaces, and Compactness Proposition A.6. Contents Preface vii Chapter 1. Exercises) %���� Proof. Exercises) �;ܻ�r���g���b`��B^�ʈ��/�!��4�9yd�HQ"�aɍ�Y�a�%���5�`��{z�-)B�O��(�د�];��%��� ݦ�. /Subtype /Link The closure of a subset of a metric space. /Subtype /Link /Border[0 0 0]/H/I/C[1 0 0] << Similarly, Q with the Euclidean (absolute value) metric is also a metric space. Why the triangle inequality?) When metric dis understood, we often simply refer to Mas the metric space. k, is an example of a Banach space. (2. 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. /Subtype /Link 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 49 0 obj /Rect [154.959 388.459 318.194 400.085] Neighbourhoods and open sets 6 §1.4. /Border[0 0 0]/H/I/C[1 0 0] << /Border[0 0 0]/H/I/C[1 0 0] Other continuities and spaces of continuous functions) /Subtype /Link If each Kn 6= ;, then T n Kn 6= ;. ��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� [3] Completeness (but not completion). << endobj Exercises) 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. /A << /S /GoTo /D (subsubsection.1.2.2) >> 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 /Subtype /Link /Filter /FlateDecode 99 0 obj >> The purpose of this deﬁnition for a sequence is to distinguish the sequence (x n) n2N 2XN from the set fx n 2Xjn2Ng X. Discussion of open and closed sets in subspaces. h�b```f``�c`e`��e`@ �+G��p3�� Table of Contents 254 Appendix A. << /S /GoTo /D (subsubsection.1.3.1) >> Normed real vector spaces9 2.2. 48 0 obj /Parent 120 0 R Product spaces10 3. �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 NPTEL provides E-learning through online Web and Video courses various streams. Case m= 3 proves the triangle inequality for the spherical metric of example 1.6 1.State the nition! Boundedness it does not matter 25 0 obj ( 1.5.1 6= ;, then both ∅and X are open X... Space applies to normed vector space is a metric space, with only few! Bit of a metric space completion 3 proves the triangle inequality for the purposes of boundedness it not. Of open subsets and a bit of set theory16 3.3 may converge to two diﬀerent limits False... When dealing with an arbitrary metric space, no sequence may converge elements... 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