If x,d is a metric space and a is a nonempty subset of x, we can make a metric d a on a by putting. Funtional analysis lecture notes for 18 mit mathematics. Ais a family of sets in cindexed by some index set a,then a o c. We do not develop their theory in detail, and we leave the veri. If x,d is a metric space we call the collection of open sets the topology induced by the metric. For all of the lecture notes, including a table of contents, download the following file pdf. Since is a complete space, the sequence has a limit. In fact, the definition for functions on r can be easily adjusted so that it applies to functions on an arbitrary metric space. The particular distance function must satisfy the following conditions. Sets endowed with a distance are called metric spaces, and they are the subject of this chapter. Click download or read online button to get metric space book now. These are actually based on the lectures delivered by prof. Introduction to real analysis fall 2014 lecture notes.
Often, if the metric dis clear from context, we will simply denote the metric space x. A metric space is a set xtogether with a metric don it, and we will use the notation x. Part ib metric and topological spaces maths lecture notes. Notes on metric spaces prakash panangaden 3rd september 2019 please ignore remarks in this font.
It is not hard to check that d is a metric on x, usually referred to as the discrete metric. Introduction let x be an arbitrary set, which could consist of vectors in rn, functions, sequences, matrices, etc. Muhammad ashfaq ex hod, department of mathematics, government college sargodha. There are many ways to make new metric spaces from old. A metric space is, essentially, a set of points together with a rule for saying how far apart two such points are. Notes of metric spaces these notes are related to section iv of b course of mathematics, paper b. A metric space is a pair x, d, where x is a set and d is a. Sep 05, 2014 the axiomatic description of a metric space is given.
Review of metric spaces hart smith department of mathematics university of washington, seattle math 524, autumn 20 hart smith math 524. I can send some notes on the exercises in sections 16 and 17 to supervisors by email. This generalization of the absolute value on ror c to the framework of vector spaces is central to modern analysis. Metric spaces, convergence of sequences, equivalent metrics, balls, open. These are the notes prepared for the course mth 304 to be o ered to undergraduate students at iit kanpur. 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. The primary aim of the book is to provide a systematic development of the theory of metric spaces of normal, upper semicontinuous fuzzy convex fuzzy sets with compact support sets, mainly on the base space. Let be a cauchy sequence in the sequence of real numbers is a cauchy sequence check it. 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. This site is like a library, use search box in the widget to get ebook that you want. These notes are collected, composed and corrected by atiq ur rehman, phd. A metric space consists of a set x together with a metric d, where x is given the metric topology induced by d. The axiomatic description of a metric space is given. An additional aim is to sketch selected applications in which these metric space results and methods are essential for a thorough.
A metric space x,d is complete if and only if every nested sequence of nonempty closed subset of x, whose diameter tends to zero, has a nonempty intersection. Metric space download ebook pdf, epub, tuebl, mobi. The proof is similar to the proof of the original banach contraction. Let be a mapping from to we say that is a limit of at, if 0 notes 1. The discussion above ensures what computer scientists call downward compatibility. We define a hausdorff topology on a fuzzy metric space introduced by kramosil and michalek kybernetica11 1975 326334 and prove some known results. This chapter will introduce the reader to the concept of metrics a class of functions which is regarded as generalization of the notion of distance and metric spaces. Characterizations of compact metric spaces france dacar, jo. In mathematics, a metric space is a set together with a metric on the set. Norms and metrics, normed vector spaces and metric spaces. Chapter 1 metric spaces these notes accompany the fall 2011 introduction to real analysis course 1. In these metric spaces notes pdf, you will study the concepts of analysis which evidently rely on the notion of distance. Some of this material is contained in optional sections of the book, but i will assume none of that and start from scratch.
Let be a mapping from to we say that is a limit of at, if 0 theory of metric spaces lecture notes and exercises. For all of the lecture notes, including a table of contents, download the following file pdf 1. Normed vector spaces and metric spaces were going to develop generalizations of the ideas of length or magnitude and distance. Completions a notcomplete metric space presents the di culty that cauchy sequences may fail to converge. Metric spaces joseph muscat2003 last revised may 2009 a revised and expanded version of these notes are now published by springer. A metric space is just a set x equipped with a function d of two variables which measures the distance between points. Informally, 3 and 4 say, respectively, that cis closed under. We begin with the familiar notions of magnitude and distance on the real line. The first part of these notes states and discusses the main results of the. Roughly speaking, a metric on the set xis just a rule to measure the distance between any two elements of x. Pdf metric spaces notes free download tutorialsduniya. General comfort with abstraction is a prerequisite.
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. Reasonably, we want to repair this situation, and in as economical way as possible. A metric space consists of a set xtogether with a function d. The lecture notes were taken by a student in the class. 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. The necessary mathematical background includes careful treatment of limits of course. A metric space is a set x where we have a notion of distance. Lecture notes analysis ii mathematics mit opencourseware. Metricandtopologicalspaces university of cambridge. We learn analysis for the rst time over the real numbers r and we take.
If x is a set with a metric, the metric topologyon x is the topology generated by the basis consisting of open balls bx. A new and even better version of toms notes is now on our web side, and we will rely on that during the whole semester. Metric spaces a metric space is a set x that has a notion of the distance dx,y between every pair of points x,y. We next give a proof of the banach contraction principle in. Metric spaces, open balls, and limit points definition. Rasmussen notes taken by dexter chua easter 2015 these notes are not endorsed by the lecturers, and i have modi ed them often. 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. The last of these conditions is known as the triangle inequality. Notice that all this distances can be written as dx,y. For the love of physics walter lewin may 16, 2011 duration. A metric space is called complete if every cauchy sequence converges to a limit. Chapter 1 metric spaces islamic university of gaza. Metric spaces are generalizations of the real line, in which some of the theorems that hold for r.