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. 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. There are many ways to make new metric spaces from old. For all of the lecture notes, including a table of contents, download the following file pdf. We define a hausdorff topology on a fuzzy metric space introduced by kramosil and michalek kybernetica11 1975 326334 and prove some known results. Metric space download ebook pdf, epub, tuebl, mobi. Sep 05, 2014 the axiomatic description of a metric space is given. Reasonably, we want to repair this situation, and in as economical way as possible. Chapter 1 metric spaces these notes accompany the fall 2011 introduction to real analysis course 1. Some of this material is contained in optional sections of the book, but i will assume none of that and start from scratch. Informally, 3 and 4 say, respectively, that cis closed under.
Click download or read online button to get metric space book now. It is not hard to check that d is a metric on x, usually referred to as the discrete metric. Characterizations of compact metric spaces france dacar, jo. Sets endowed with a distance are called metric spaces, and they are the subject of this chapter. These are actually based on the lectures delivered by prof. A metric space is, essentially, a set of points together with a rule for saying how far apart two such points are. Let be a cauchy sequence in the sequence of real numbers is a cauchy sequence check it. Metric spaces joseph muscat2003 last revised may 2009 a revised and expanded version of these notes are now published by springer. The necessary mathematical background includes careful treatment of limits of course. The axiomatic description of a metric space is given. A metric space is a set x where we have a notion of distance. These are the notes prepared for the course mth 304 to be o ered to undergraduate students at iit kanpur. These notes are collected, composed and corrected by atiq ur rehman, phd. Let be a mapping from to we say that is a limit of at, if 0 theory of metric spaces lecture notes and exercises.
Norms and metrics, normed vector spaces and metric spaces. Often, if the metric dis clear from context, we will simply denote the metric space x. Pdf metric spaces notes free download tutorialsduniya. This site is like a library, use search box in the widget to get ebook that you want. Notes on metric spaces prakash panangaden 3rd september 2019 please ignore remarks in this font. Funtional analysis lecture notes for 18 mit mathematics. In these metric spaces notes pdf, you will study the concepts of analysis which evidently rely on the notion of distance. The discussion above ensures what computer scientists call downward compatibility. Completions a notcomplete metric space presents the di culty that cauchy sequences may fail to converge. Notes on metric spaces 2 thisisnottheonlydistancewecouldde. 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.
We do not develop their theory in detail, and we leave the veri. Review of metric spaces hart smith department of mathematics university of washington, seattle math 524, autumn 20 hart smith math 524. Rasmussen notes taken by dexter chua easter 2015 these notes are not endorsed by the lecturers, and i have modi ed them often. Notes of metric spaces these notes are related to section iv of b course of mathematics, paper b. Metric spaces, convergence of sequences, equivalent metrics, balls, open. A metric space is a set xtogether with a metric don it, and we will use the notation x.
A metric space consists of a set x together with a metric d, where x is given the metric topology induced by d. 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. The last of these conditions is known as the triangle inequality. A metric space is just a set x equipped with a function d of two variables which measures the distance between points. General comfort with abstraction is a prerequisite. For all of the lecture notes, including a table of contents, download the following file pdf 1. Lecture notes analysis ii mathematics mit opencourseware. If x,d is a metric space we call the collection of open sets the topology induced by the metric. We next give a proof of the banach contraction principle in.
We begin with the familiar notions of magnitude and distance on the real line. If x is a set with a metric, the metric topologyon x is the topology generated by the basis consisting of open balls bx. Introduction let x be an arbitrary set, which could consist of vectors in rn. Let be a mapping from to we say that is a limit of at, if 0 notes 1. For the love of physics walter lewin may 16, 2011 duration. Metricandtopologicalspaces university of cambridge. A metric space is called complete if every cauchy sequence converges to a limit. I can send some notes on the exercises in sections 16 and 17 to supervisors by email. A metric space is, essentially, a set of points together with a rule for. 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.
The lecture notes were taken by a student in the class. 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. A metric space is a set x together with a function d. We learn analysis for the rst time over the real numbers r and we take. Chapter 1 metric spaces islamic university of gaza. The particular distance function must satisfy the following conditions. Lecture notes assignments download course materials. In mathematics, a metric space is a set together with a metric on the set.
An additional aim is to sketch selected applications in which these metric space results and methods are essential for a thorough. Part ib metric and topological spaces based on lectures by j. Normed vector spaces and metric spaces were going to develop generalizations of the ideas of length or magnitude and distance. Well generalize from euclidean spaces to more general spaces, such as spaces of functions. 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. The first part of these notes states and discusses the main results of the. A metric space consists of a set xtogether with a function d. 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 generalization of the absolute value on ror c to the framework of vector spaces is central to modern analysis. Introduction let x be an arbitrary set, which could consist of vectors in rn, functions, sequences, matrices, etc.
Summer 2007 john douglas moore our goal of these notes is to explain a few facts regarding metric spaces not. Roughly speaking, a metric on the set xis just a rule to measure the distance between any two elements of x. Part ib metric and topological spaces maths lecture notes. The proof is similar to the proof of the original banach contraction. Muhammad ashfaq ex hod, department of mathematics, government college sargodha. A metric space is a pair x, d, where x is a set and d is a. Ais a family of sets in cindexed by some index set a,then a o c. 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. Metric spaces, open balls, and limit points definition. Notes on metric spaces these notes are an alternative to the textbook, from and including closed sets and open sets page 58 to and excluding cantor sets page 95 1 the topology of metric spaces assume m is a metric space with distance function d. 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.