# metric space pdf notes

0:We write NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. Metric Spaces A metric space is a set X endowed with a metric ρ : X × X → [0,∞) that satisﬁes the following properties for all x, y, and z in X: 1. ρ(x,y) = 0 if and only if x = y, 2. ρ(x,y) = ρ(y,x), and 3. ρ(x,z) ≤ ρ(x,y)+ ρ(y,z). Notes of Metric Spaces These notes are related to Section IV of B Course of Mathematics, paper B. These notes are collected, composed and corrected by Atiq ur Rehman, PhD. %PDF-1.5 This distance function is known as the metric. A metric space (X;d) is a … In these “Metric Spaces Notes PDF”, we will study the concepts of analysis which evidently rely on the notion of distance. It helps to have a unifying framework for discussing both random variables and stochastic processes, as well as their convergence, and such a framework is provided by metric spaces. We call the‘8 taxicab metric on (‘8Þ For , distances are measured as if you had to move along a rectangular grid of8œ# city streets from to the taxicab cannot cut diagonally across a city blockBC ). Students can easily make use of all these Metric Spaces Notes PDF by downloading them. 94 7. In this course, the objective is to develop the usual idea of distance into an abstract form on any set of objects, maintaining its inherent characteristics, and the resulting consequences. endobj We are very thankful to Mr. Tahir Aziz for sending these notes. B r(x) is the standard ball of radius rcentered at xand B1 r (x) is the cube of length rcentered at x. Notes on Metric Spaces These notes introduce the concept of a metric space, which will be an essential notion throughout this course and in others that follow. Ark1: Metric spaces MAT2400 — spring 2012 Subset metrics Problem 24. METRIC SPACES AND SOME BASIC TOPOLOGY De¿nition 3.1.2 Real n-space,denotedUn, is the set all ordered n-tuples of real numbers˚ i.e., Un x1˛x2˛˝˝˝˛xn : x1˛x2˛˝˝˝˛xn + U . ?�ྍ�ͅ�伣M�0Rk��PFv*�V�����d֫V��O�~��� De nition (Metric space). A metric space is a non-empty set equi pped with structure determined by a well-defin ed notion of distan ce. The deﬁnition of a metric Deﬁnition – Metric A metric on a set X is a function d that assigns a real number to each pair of elements of X in such a way that the following properties hold. Continuity & Uniform Continuity in Metric Spaces: Continuous mappings, Sequential criterion and other characterizations of continuity, Uniform continuity, Homeomorphism, Contraction mapping, Banach fixed point theorem. Since is a complete space, the sequence has a limit. Lecture Notes on Metric Spaces Math 117: Summer 2007 John Douglas Moore Our goal of these notes is to explain a few facts regarding metric spaces not included in the ﬁrst few chapters of the text , in the hopes of providing an easier transition to more advanced texts such as . De nitions, and open sets. The topics we will cover in these Metric Spaces Notes PDF will be taken from the following list: Basic Concepts: Metric spaces: Definition and examples, Sequences in metric spaces, Cauchy sequences, Complete metric space. Metric Spaces The following de nition introduces the most central concept in the course. Source: daiict.ac.in, Metric Spaces Handwritten Notes The term ‘m etric’ i s d erived from the word metor (measur e). Therefore our de nition of a complete metric space applies to normed vector spaces: an n.v.s. 3 0 obj <>/Font<>/XObject<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 612 792] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>> (This is problem 2.47 in the book) Proof. We have provided multiple complete Metric Spaces Notes PDF for any university student of BCA, MCA, B.Sc, B.Tech CSE, M.Tech branch to enhance more knowledge about the subject and to score better marks in the exam. <>>> 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 . These are not the same thing. 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. Source: iitk.ac.in, Metric Spaces Notes Metric Spaces Notes PDF. called a discrete metric; (X;d) is called a discrete metric space. Let ϵ>0 be given. We will write (X,ρ) to denote the metric space X endowed with a metric ρ. 1 The dot product If x = (x Topology of Metric Spaces 1 2. Example 7.4. 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. A function f: X!Y is said to be continuous if for any Uopen in Y, f 1(U) is open in X. Theorem 1.6.2 Let X, Y be topological spaces, and f: X!Y, then TFAE: Notes of Metric Space Level: BSc or BS, Author: Umer Asghar Available online @ , Version: 1.0 METRIC SPACE:-Let be a non-empty set and denotes the set of real numbers. 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 Then there is an automatic metric d Y on Y deﬁned by restricting dto the subspace Y× Y, d Y = dY| × Y. If xn! Proposition. NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. 4 0 obj By the deﬁnition of convergence, 9N such that d„xn;x” <ϵ for all n N. fn 2 N: n Ng is inﬁnite, so x is an accumulation point. endobj spaces and σ-ﬁeld structures become quite complex. In this course, the objective is to develop the usual idea of distance into an abstract form on any set of objects, maintaining its inherent characteristics, and the resulting consequences. View Notes - notes_on_metric_spaces_0.pdf from MATH 321 at Maseno University. In nitude of Prime Numbers 6 5. Deﬁnition 1. Source: spcmc.ac.in, Metric Spaces Handwritten Notes 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. Therefore ‘1is a normed vector space. Let (X,d) denote a metric space, and let A⊆X be a subset. ���A��..�O�b]U*� ���7�:+�v�M}Y�����p]_�����.�y �i47ҨJ��T����+�3�K��ʊPD� m�n��3�EwB�:�ۓ�7d�J:��'/�f�|�r&�Q ���Q(��V��w��A�0wGQ�2�����8����S`Gw�ʒ�������r���@T�A��G}��}v(D.cvf��R�c�'���)(�9����_N�����O����*xDo�N�ׁ�rw)0�ϒ�(�8�a�I}5]�Q�sV�2T�9W/\�Y}��#�1\�6���Hod�a+S�ȍ�r-��z�s���. Suppose x′ is another accumulation point. The ﬁrst goal of this course is then to deﬁne metric spaces and continuous functions between metric spaces. Proof. stream x, then x is the only accumulation point of fxng1 n 1 Proof. Given a metric don X, the pair (X,d) is called a metric space. Suppose that Mis a compact metric space and that SˆMis a closed subspace. Metric Spaces (Notes) These are updated version of previous notes. Proof. It is easy to check that satisfies properties .Ðß.Ñ .>> >1)-5) so is a metric space. A metric space is called complete if every Cauchy sequence converges to a limit. Suppose {x n} is a convergent sequence which converges to two diﬀerent limits x 6= y. In other words, no sequence may converge to two diﬀerent limits. 1 0 obj 1.1 Metric Space 1.1-1 Definition. (M3) d( x, y ) = d( y, x ). 1 An \Evolution Variational Inequality" on a metric space The aim of this section is to introduce an evolution variational inequality (EVI) on a metric space which will be the main subject of these notes. Topological Spaces 3 3. The third property is called the triangle inequality. If a metric space Xis not complete, one can construct its completion Xb as follows. 2 0 obj Introduction Let X … Connectedness and Compactness: Connectedness, Connected subsets of R, Connectedness and continuous mappings, Compactness, Compactness and boundedness, Continuous functions on compact spaces. Metric spaces Lecture notes for MA2223 P. Karageorgis pete@maths.tcd.ie 1/20. Deﬁnition 1.2.1. Many mistakes and errors have been removed. Metric Spaces Handwritten Notes Basis for a Topology 4 4. The same set can be … … A useful metric on this space is the tree metric, d(x,y) = 1 min{n: xn ̸= yn}. is complete if it’s complete as a metric space, i.e., if all Cauchy sequences converge to elements of the n.v.s. A metric space X is called a complete metric space if every Cauchy sequence in X converges to some point in X. The limit of a sequence in a metric space is unique. Topology Generated by a Basis 4 4.1. Introduction Let X be an arbitrary set, which could consist of vectors in Rn, functions, sequences, matrices, etc. We have listed the best Metric Spaces Reference Books that can help in your Metric Spaces exam preparation: Student Login for Download Admit Card for OBE Examination, Step-by-Step Guide for using the DU Portal for Open-Book Examination (OBE), Open Book Examination (OBE) for the final semester/term/year students, Computer Algebra Systems & Related Software Notes, Introduction to Information Theory & Coding Notes, Mathematical Modeling & Graph Theory Notes, Riemann Integration & Series of Functions Notes. 2 Open balls and neighborhoods Let (X,d) be a metric space… We can easily convert our de nition of bounded sequences in a normed vector space into a de nition of bounded sets and bounded functions. Topology of Metric Spaces: Open and closed ball, Neighborhood, Open set, Interior of a set, Limit point of a set, Derived set, Closed set, Closure of a set, Diameter of a set, Cantor’s theorem, Subspaces, Dense set. Let X be a metric space. Analysis on metric spaces 1.1. Think of the plane with its usual distance function as you read the de nition. The purpose of this deﬁnition for a sequence is to distinguish the sequence (x n) n2N 2XN from the set fx n 2Xjn2Ng X. 1.6 Continuous functions De nition 1.6.1 Let X, Y be topological spaces. <> §1. 1 Metric spaces IB Metric and Topological Spaces 1 Metric spaces 1.1 De nitions As mentioned in the introduction, given a set X, it is often helpful to have a notion of distance between points. d(f,g) is not a metric in the given space. Metric Spaces, Topological Spaces, and Compactness sequences in X;where we say (x ) ˘ (y ) provided d(x ;y ) ! Source: math.iitb.ac.in, Metric Spaces Notes The second is the set that contains the terms of the sequence, and if Already know: with the usual metric is a complete space. Deﬁnition. (Tom’s notes 2.3, Problem 33 (page 8 and 9)). These are the notes prepared for the course MTH 304 to be o ered to undergraduate students at IIT Kanpur. A metric space is a pair ( X, d ), where X is a set and d is a metric on X; that is a function on X X such that for all x, y, z X, we have: (M1) d( x, y ) 0. (M4) d( x, y ) d( x, z ) + d( z, y ). Metric Spaces Then d is a metric on R. Nearly all the concepts we discuss for metric spaces are natural generalizations of the corresponding concepts for R with this absolute-value metric. Thus, Un U_ ˘U˘ ˘^] U‘ nofthem, the Cartesian product of U with itself n times. In these “Metric Spaces Notes PDF”, we will study the concepts of analysis which evidently rely on the notion of distance. De nition 1.1. METRIC SPACES, TOPOLOGY, AND CONTINUITY Lemma 1.1. This distance function TOPOLOGY: NOTES AND PROBLEMS Abstract. %���� NOTES FOR MATH 4510, FALL 2010 DOMINGO TOLEDO 1. 74 CHAPTER 3. Let be a Cauchy sequence in the sequence of real numbers is a Cauchy sequence (check it!). 4 ALEX GONZALEZ A note of waning! MAT 314 LECTURE NOTES 1. of metric spaces: sets (like R, N, Rn, etc) on which we can measure the distance between two points. 2.1. METRIC SPACES 5 Remark 1.1.5. We motivate the de nition by means of two examples. <> x��]ms�F����7����˻�o�is��䮗i�A��3~I%�m���%e�\$d��N]��,�X,��ŗ?O�~�����BϏ��/�z�����.t�����^�e0E4�Ԯp66�*�����/��l��������W�{��{��W�|{T�F�����A�hMi�Q_�X�P����_W�{�_�]]V�x��ņ��XV�t§__�����~�|;_-������O>Φnr:���r�k��_�{'�?��=~��œbj'��A̯ And by replacing the norm in the de nition with the distance function in a metric space, we can extend these de nitions from normed vector spaces to general metric spaces. Remark 3.1.3 From MAT108, recall the de¿nition of an ordered pair: a˛b def Product Topology 6 6. Let an element ˘of Xb consist of an equivalence class of Cauchy 251. Bounded sets in metric spaces. Source: princeton.edu. Still, you should check the �?��No~� ��*�R��_�įsw\$��}4��=�G�T�y�5P��g�:҃l. Metric Space (Handwritten Classroom Study Material) Submitted by Sarojini Mohapatra (MSc Math Student) Central University of Jharkhand ... P Kalika Notes (Provide your Feedbacks/Comments at maths.whisperer@gmail.com) Title: Metric Space Notes Author: P Kalika Subject: Metric Space (M2) d( x, y ) = 0 if and only if x = y. Suppose dis a metric on Xand that Y ⊆ X. Let (X;d) be a metric space and let A X. Deﬁnition. Sequence ( check it! ) numbers is a … View notes - notes_on_metric_spaces_0.pdf from MATH 321 at Maseno.. Complete if it ’ s complete as a metric space is a convergent which! A non-empty set equi pped with structure determined by a well-defin ed notion of ce. With itself n times for MATH 4510, FALL 2010 DOMINGO TOLEDO 1 you read the de nition let... 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This is Problem 2.47 in the book, but I will assume none of and. From the word metor ( measur e ) can easily make use of all these metric spaces —! From scratch we will study the concepts of analysis which evidently rely on the notion of distance Together with,... Spaces notes PDF by downloading them Cauchy 251 read the de nition means. Spaces the following de nition 1.6.1 let x be an arbitrary set, could... Central concept in the book, but I will assume none of that and start from scratch most central in! Limit of a compact metric space is called complete if it ’ s notes 2.3 Problem. > 1 ) -5 ) so is a non-empty set equi pped with structure determined by a well-defin notion... Mathematics, paper B to a limit ; x0 ) = d ( z, y ) 0!, z ) + d ( z, y ) vectors in Rn functions... Space ( y, x ) ρ ) to denote the metric (., with the usual metric is a … View notes - notes_on_metric_spaces_0.pdf from MATH 321 at University. Is complete if it ’ s notes 2.3, Problem 33 ( 8!