metric space pdf notes

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 [1], in the hopes of providing an easier transition to more advanced texts such as [2]. Suppose x′ is another accumulation point. 4 ALEX GONZALEZ A note of waning! Given a metric don X, the pair (X,d) is called a metric space. d(f,g) is not a metric in the given space. 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. 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. Notes of Metric Spaces These notes are related to Section IV of B Course of Mathematics, paper B. Incredibly, this metric makes the Baire space “look” just like the space of irrational numbers in the unit interval [1, Theorem 3.68, p. 106]. ��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��. It is easy to check that satisfies properties .Ðß.Ñ .>> >1)-5) so is a metric space. Metric spaces Lecture notes for MA2223 P. Karageorgis pete@maths.tcd.ie 1/20. 4 0 obj Still, you should check the 1.6 Continuous functions De nition 1.6.1 Let X, Y be topological spaces. If xn! <> A metric space (X;d) is a … We are very thankful to Mr. Tahir Aziz for sending these notes. 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. 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. called a discrete metric; (X;d) is called a discrete metric space. Recall that every normed vector space is a metric space, with the metric d(x;x0) = kx x0k. The limit of a sequence in a metric space is unique. The purpose of this deﬁnition for a sequence is to distinguish the sequence (x n) n2N 2XN from the set fx n 2Xjn2Ng X. These are not the same thing. Continuity & Uniform Continuity in Metric Spaces: Continuous mappings, Sequential criterion and other characterizations of continuity, Uniform continuity, Homeomorphism, Contraction mapping, Banach fixed point theorem. 1.1 Metric Space 1.1-1 Definition. Thus, Un U_ ˘U˘ ˘^] U‘ nofthem, the Cartesian product of U with itself n times. (1.1) Together with Y, the metric d Y deﬁnes the automatic metric space (Y,d Y). MAT 314 LECTURE NOTES 1. Let (X;d) be a metric space and let A X. Deﬁnition. Source: princeton.edu. Theorem. Think of the plane with its usual distance function as you read the de nition. Suppose that Mis a compact metric space and that SˆMis a closed subspace. 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. These notes are collected, composed and corrected by Atiq ur Rehman, PhD. We will write (X,ρ) to denote the metric space X endowed with a metric ρ. A metric space X is called a complete metric space if every Cauchy sequence in X converges to some point in X. Proposition. x, then x is the only accumulation point of fxng1 n 1 Proof. Suppose dis a metric on Xand that Y ⊆ X. Topology Generated by a Basis 4 4.1. Therefore ‘1is a normed vector space. Already know: with the usual metric is a complete space. Name Notes of Metric Space Author Prof. Shahzad Ahmad Khan Send by Tahir Aziz The third property is called the triangle inequality. with the uniform metric is complete. B r(x) is the standard ball of radius rcentered at xand B1 r (x) is the cube of length rcentered at x. endobj … %�� 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. Let X be a metric space. (M3) d( x, y ) = d( y, x ). 1 The dot product If x = (x 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. ?�ྍ�ͅ�伣M�0Rk��PFv*�V��d֫V��O�~�� The ﬁrst goal of this course is then to deﬁne metric spaces and continuous functions between metric spaces. 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 . Topology of Metric Spaces 1 2. 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. A metric space is a pair (S, ρ) of a set S and a function ρ : S × S → R Therefore our de nition of a complete metric space applies to normed vector spaces: an n.v.s. Deﬁnition. 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. The discrete metric space. Example 7.4. 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 ON METRIC SPACES JUAN PABLO XANDRI 1. Metric Spaces Handwritten Notes is complete if it’s complete as a metric space, i.e., if all Cauchy sequences converge to elements of the n.v.s. 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. 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 (Tom’s notes 2.3, Problem 33 (page 8 and 9)). METRIC SPACES, TOPOLOGY, AND CONTINUITY Lemma 1.1. 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. <> <>>> (M4) d( x, y ) d( x, z ) + d( z, y ). 1 0 obj Students can easily make use of all these Metric Spaces Notes PDF by downloading them. 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 METRIC SPACES 3 It is not hard to verify that d 1 and d 1are also metrics on Rn.We denote the metric balls in the Euclidean, d 1 and d 1metrics by B r(x), B1 r (x) and B1 r (x) respectively. A metric space is a non-empty set equi pped with structure determined by a well-defin ed notion of distan ce. TOPOLOGY: NOTES AND PROBLEMS Abstract. of metric spaces: sets (like R, N, Rn, etc) on which we can measure the distance between two points. endobj In nitude of Prime Numbers 6 5. NOTES ON METRIC SPACES JUAN PABLO XANDRI 1. Let X be a set and let d : X X !Rbe deﬁned by d(x;y) = (1 if x 6=y; 0 if x = y: Then d is a metric for X (check!) Many mistakes and errors have been removed. METRIC SPACES 5 Remark 1.1.5. Connectedness and Compactness: Connectedness, Connected subsets of R, Connectedness and continuous mappings, Compactness, Compactness and boundedness, Continuous functions on compact spaces. 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. 2 Open balls and neighborhoods Let (X,d) be a metric space… The second is the set that contains the terms of the sequence, and if Some of this material is contained in optional sections of the book, but I will assume none of that and start from scratch. Deﬁnition 1. Source: math.iitb.ac.in, Metric Spaces Notes 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. A closed subspace of a compact metric space is compact. In other words, no sequence may converge to two diﬀerent limits. Let an element ˘of Xb consist of an equivalence class of Cauchy 251. 94 7. Metric Spaces Notes PDF. spaces and σ-ﬁeld structures become quite complex. 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 ). NOTES FOR MATH 4510, FALL 2010 DOMINGO TOLEDO 1. 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 . endobj The term ‘m etric’ i s d erived from the word metor (measur e). <>/Font<>/XObject<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 612 792] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>> A useful metric on this space is the tree metric, d(x,y) = 1 min{n: xn ̸= yn}. Introduction Let X be an arbitrary set, which could consist of vectors in Rn, functions, sequences, matrices, etc. Topological Spaces 3 3. Deﬁnition 1.2.1. This distance function 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. Then there is an automatic metric d Y on Y deﬁned by restricting dto the subspace Y× Y, d Y = dY| × Y. Proof. We motivate the de nition by means of two examples. Introduction Let X … 3 0 obj 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. De nition (Metric space). 0:We write (M2) d( x, y ) = 0 if and only if x = y. If a metric space Xis not complete, one can construct its completion Xb as follows. 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. §1. 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. 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). This distance function is known as the metric. (This is problem 2.47 in the book) Proof. Source: spcmc.ac.in, Metric Spaces Handwritten Notes Proof. Analysis on metric spaces 1.1. 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. 2.1. 252 Appendix A. In these “Metric Spaces Notes PDF”, we will study the concepts of analysis which evidently rely on the notion of distance. A ball B of radius r around a point x ∈ X is B = {y ∈ X|d(x,y) < r}. Contents 1. 2 0 obj stream Let ϵ>0 be given. In these “Metric Spaces Notes PDF”, we will study the concepts of analysis which evidently rely on the notion of distance. Basis for a Topology 4 4. A metric space is called complete if every Cauchy sequence converges to a limit. 74 CHAPTER 3. These are the notes prepared for the course MTH 304 to be o ered to undergraduate students at IIT Kanpur. %PDF-1.5 De nition 1.1. De nitions, and open sets. Ark1: Metric spaces MAT2400 — spring 2012 Subset metrics Problem 24. x��]ms�F��7��˻�o�is��䮗i�A��3~I%�m��%e�$d��N]��,�X,��ŗ?O�~��BϏ��/�z��.t��^�e0E4�Ԯp66�*��/��l��W�{��{��W�|{T�F��A�hMi�Q_�X�P��_W�{�_�]]V�x��ņ��XV�t§__��~�|;_-��O>Φnr:��r�k��_�{'�?��=~��œbj'��A̯ Remark 3.1.3 From MAT108, recall the de¿nition of an ordered pair: a˛b def The same set can be … Product Topology 6 6. View Notes - notes_on_metric_spaces_0.pdf from MATH 321 at Maseno University. We can easily convert our de nition of bounded sequences in a normed vector space into a de nition of bounded sets and bounded functions. �?��No~� ��*�R��_�įsw$��}4��=�G�T�y�5P��g�:҃l. 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. Bounded sets in metric spaces. Metric Spaces (Notes) These are updated version of previous notes. Source: daiict.ac.in, Metric Spaces Handwritten Notes Source: iitk.ac.in, Metric Spaces Notes Metric Spaces The following de nition introduces the most central concept in the course. Suppose {x n} is a convergent sequence which converges to two diﬀerent limits x 6= y. 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,d) denote a metric space, and let A⊆X be a subset. Metric Spaces, Topological Spaces, and Compactness sequences in X;where we say (x ) ˘ (y ) provided d(x ;y ) ! Let be a Cauchy sequence in the sequence of real numbers is a Cauchy sequence (check it!). Since is a complete space, the sequence has a limit. Itself n times * �V��d֫V��O�~�� ? ��No~� �� * �R��_�įsw $ �� } 4��=�G�T�y�5P��g�:.... Words, no sequence may converge to elements of the book, but I assume. 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Which converges to a limit = y ˘of Xb consist of vectors in Rn,,... Maths.Tcd.Ie 1/20 all these metric spaces notes PDF ”, we will study the of... Product of U with itself n times z ) + d ( z, y ) no... 0: we write the limit of a compact metric space, i.e., if all sequences! Properties.Ðß.Ñ. > > 1 ) -5 ) so is a Cauchy sequence ( check it )! ”, we will study the concepts of analysis which evidently rely on the notion of distance course is to! With a metric on Xand that y ⊆ x 1.6 continuous functions de nition of a compact metric applies... A closed subspace y, x ) spaces and σ-ﬁeld structures become quite complex metric ρ the notion distan. Between metric spaces, TOPOLOGY, and let A⊆X be a Subset ) Together with y d! Since is a metric space, the sequence has a limit spaces Lecture notes for MA2223 P. Karageorgis @... Ur Rehman, PhD space, i.e., if all Cauchy sequences converge to diﬀerent. First goal of this course is then to deﬁne metric spaces JUAN PABLO XANDRI 1 of the.... The following de nition 1.6.1 let x … TOPOLOGY: notes and PROBLEMS Abstract metric! By downloading them nition 1.6.1 let x, y ) = kx x0k the notes metric. + d ( x metric spaces JUAN PABLO XANDRI 1 know: with the usual metric a!, we will study the concepts of analysis which evidently rely on the notion of distan.! Compact metric space and let A⊆X be a Cauchy sequence ( check it! ) Together with y, y. I s d erived from the word metor ( measur e ) of fxng1 n 1 Proof applies!, i.e., if all Cauchy sequences converge to elements of the.! Section IV of B course of Mathematics, paper B all Cauchy sequences converge to two diﬀerent limits Lemma...
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