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