Metric spaces notes pdf

We begin with the familiar notions of magnitude and distance on the real line. Metric spaces, open balls, and limit points definition. For the love of physics walter lewin may 16, 2011 duration. We next give a proof of the banach contraction principle in. General comfort with abstraction is a prerequisite. Often, if the metric dis clear from context, we will simply denote the metric space x. 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. 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. Muhammad ashfaq ex hod, department of mathematics, government college sargodha. This generalization of the absolute value on ror c to the framework of vector spaces is central to modern analysis.

In these metric spaces notes pdf, you will study the concepts of analysis which evidently rely on the notion of distance. 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 lecture notes were taken by a student in the class. A metric space is called complete if every cauchy sequence converges to a limit. Completions a notcomplete metric space presents the di culty that cauchy sequences may fail to converge. It is not hard to check that d is a metric on x, usually referred to as the discrete metric. The first part of these notes states and discusses the main results of the. Review of metric spaces hart smith department of mathematics university of washington, seattle math 524, autumn 20 hart smith math 524. Notes on metric spaces 2 thisisnottheonlydistancewecouldde. Metric spaces are generalizations of the real line, in which some of the theorems that hold for r. 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 necessary mathematical background includes careful treatment of limits of course. Metric spaces joseph muscat2003 last revised may 2009 a revised and expanded version of these notes are now published by springer. The last of these conditions is known as the triangle inequality. Introduction let x be an arbitrary set, which could consist of vectors in rn. We learn analysis for the rst time over the real numbers r and we take. Let be a mapping from to we say that is a limit of at, if 0 theory of metric spaces lecture notes and exercises. 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. If x is a set with a metric, the metric topologyon x is the topology generated by the basis consisting of open balls bx. The axiomatic description of a metric space is given. Part ib metric and topological spaces maths lecture notes. Part ib metric and topological spaces based on lectures by j. I can send some notes on the exercises in sections 16 and 17 to supervisors by email. Since is a complete space, the sequence has a limit. Notice that all this distances can be written as dx,y.

Metric spaces, convergence of sequences, equivalent metrics, balls, open. Reasonably, we want to repair this situation, and in as economical way as possible. Lecture notes analysis ii mathematics mit opencourseware. Chapter 1 metric spaces islamic university of gaza. Pdf metric spaces notes free download tutorialsduniya.

Chapter 1 metric spaces these notes accompany the fall 2011 introduction to real analysis course 1. These are the notes prepared for the course mth 304 to be o ered to undergraduate students at iit kanpur. An additional aim is to sketch selected applications in which these metric space results and methods are essential for a thorough. A metric space is, essentially, a set of points together with a rule for. Let be a mapping from to we say that is a limit of at, if 0 notes 1. Roughly speaking, a metric on the set xis just a rule to measure the distance between any two elements of x. 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. 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. Introduction to real analysis fall 2014 lecture notes. Ais a family of sets in cindexed by some index set a,then a o c. For all of the lecture notes, including a table of contents, download the following file pdf.

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. Well generalize from euclidean spaces to more general spaces, such as spaces of functions. A metric space is a set x together with a function d. 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. A metric space is a set x where we have a notion of distance. This site is like a library, use search box in the widget to get ebook that you want. The discussion above ensures what computer scientists call downward compatibility. Some of this material is contained in optional sections of the book, but i will assume none of that and start from scratch. Notes of metric spaces these notes are related to section iv of b course of mathematics, paper b. Metricandtopologicalspaces university of cambridge. 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. In mathematics, a metric space is a set together with a metric on the set.

A metric space is just a set x equipped with a function d of two variables which measures the distance between points. Metric space download ebook pdf, epub, tuebl, mobi. Rasmussen notes taken by dexter chua easter 2015 these notes are not endorsed by the lecturers, and i have modi ed them often. Lecture notes assignments download course materials. Notes on metric spaces prakash panangaden 3rd september 2019 please ignore remarks in this font.

Let be a cauchy sequence in the sequence of real numbers is a cauchy sequence check it. If x,d is a metric space we call the collection of open sets the topology induced by the metric. Summer 2007 john douglas moore our goal of these notes is to explain a few facts regarding metric spaces not. 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. These notes are collected, composed and corrected by atiq ur rehman, phd. 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. Informally, 3 and 4 say, respectively, that cis closed under. Click download or read online button to get metric space book now.

A metric space consists of a set x together with a metric d, where x is given the metric topology induced by d. Funtional analysis lecture notes for 18 mit mathematics. Normed vector spaces and metric spaces were going to develop generalizations of the ideas of length or magnitude and distance. A metric space consists of a set xtogether with a function d. We do not develop their theory in detail, and we leave the veri. In fact, the definition for functions on r can be easily adjusted so that it applies to functions on an arbitrary metric space.

The proof is similar to the proof of the original banach contraction. For all of the lecture notes, including a table of contents, download the following file pdf 1. 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. Introduction let x be an arbitrary set, which could consist of vectors in rn, functions, sequences, matrices, etc.