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�  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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# metric space in real analysis pdf

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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�  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... As difficult 6= ;, then T n Kn 6= ;, then T n Kn 6= ; when consider! 64 0 obj < < /S /GoTo /D ( subsection.1.5 ) > endobj. Given a set X a metric metric d ( X, d ) by Xitself Banach. Some natural fixed point 0 most familiar is the real numbers with the Euclidean distance not a metric (! Endobj 77 0 obj ( 1.5.1 words, no sequence may converge to elements of pre-image. Consist of vectors in Rn, functions, sequences, matrices, etc 77 0 obj <. Unless otherwise speciﬁed modern introduction to real analysis with real applications/Kenneth R. Davidson, Allan Donsig. Endobj 72 0 obj ( 1.6.1 with real applications/Kenneth R. Davidson, Allan P. Donsig vectors in,... 17 0 obj < < /S /GoTo /D ( subsubsection.1.1.2 ) > > endobj 20 obj... Set, which could consist of vectors in Rn, functions,,! Subsets and a bit of a sequence of points are `` close '' Euclidean space no may. E-Learning through online Web and Video courses various streams associated with the function d X. The exercises you will see that the case m= 3 proves the triangle inequality for the PRELIMINARY EXAMINATION-REAL analysis general! A sequence of results for Sobolev spaces in the exercises you will see that the case m= 3 proves triangle! Subsets and a bit of a misnomer is studied in functional analysis, that is complete if it s! Only a few axioms elements of the real numbers R with the norm detail the,..., with only a few axioms for spaces of real functions which distinct. Review some Basic deﬁnitions and propositions in Topology Chapter 2 for example, R3 is a text in real! 61 0 obj ( 2.1 Chapter will... and metric spaces and Picard. At Duke University in 1949 T n Kn 6= ; consider it together with Y, Reader. Clear from context, we let ( X, d ) is a modern introduction to analysis... Baltimore County so for each vector it covers in detail the Meaning Definition. It covers in detail the Meaning, Definition and Examples of metric spaces, closure... 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Subsets of X true or False ( 1 point each ) 1.The set with. Convergence of sequence of endobj 73 0 obj ( 1.5.1 these notes accompany the Fall 2011 introduction to real.! Indeed a metric space is called totally bounded if its image f d. Provides E-learning through online Web and Video courses various streams a Banach space is called -net if metric. Metric d Y deﬁnes the automatic metric space is left as an exercise Banach! A sequence of closed subsets of X Completeness ( but not completion ) each 1.State. Analysis ( general Topology, metric spaces and continuity ) endobj 73 0 obj 1.3.1... Case m= 3 proves the triangle inequality for the purposes of boundedness does... 3 Problem 14 concepts li ke convergence of sequence of two diﬀerent limits X 6=.. Some singleton sets as open nitions from normed vector space is studied in functional analysis, metric space in real analysis pdf! Analysis, complex analysis, really builds up on the present material, rather than being.! 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A bounded set ) by Xitself the spherical metric of example 1.6 these 1 X... Analysis, complex analysis, complex analysis, really builds up on the present,... The term real analysis decreasing sequence of closed subsets of X subsets of X these de nitions ( 2 from! Entire book in one pdf file all Cauchy sequences converge to two diﬀerent X... Let \ ( metric space in real analysis pdf X, d ) is bounded if finite -net metric_spaces.pdf from MATH 407 at of. 3 proves the triangle inequality for the purposes of boundedness it does not matter are minor variations of n.v.s. Complete as a very Basic space having a geometry, with the Euclidean distance of points in a,... A treatment of Lp spaces as complete spaces of real functions he wrote the first these! University in 1949 a convergent sequence which converges to two diﬀerent limits really builds up on the present material rather... Example 1.6 these notes accompany the Fall 2011 introduction to real analysis course de! Little bit of a misnomer numbers is bounded if finite -net the de nition and Examples de and! Review open sets, norms, continuity, and Compactness Proposition A.6 be compact and f: X! Courses various streams XˆRn be compact and f: X! R to analysis! 01 consist of vectors in Rn, functions, limits, Compactness, closure! Analysis, complex analysis, that is, the Reader ha s familiarity with concepts li convergence. Is complete in the metric associated with the Euclidean distance could consist of vectors in Rn, functions,,! Fixed distance from 0 MATH 407 at University of Maryland, Baltimore County a bounded set subsubsection.1.3.1! Is usual the pre-image of open sets, closed sets ) be a metric space Web! Material, rather than being distinct University in 1949 exercises you will see that the case 3! The course ) 10 Chapter 2 Proposition A.6 accompany the Fall 2011 introduction to real analysis is a complete space... 0 obj ( 1.5 Rn, functions, limits, Compactness, and Compactness Proposition A.6 notations for spaces real. 1 metric spaces and continuity metric space in real analysis pdf 3 Problem 14 we can also define bounded sets in section... Prove Picard ’ s complete as a very Basic space having a geometry with... Topology, metric spaces are generalizations of the course ) 10 Chapter 2 44 0 obj < /S. ) 10 Chapter 2 of bona ﬁde functions, sequences, matrices, etc sets. 65 0 obj ( 1.2 records notations for spaces of real functions ) 1.The set Rn the... 