Metric spaces are generalizations of the real line, in which some of the theorems that hold for r remain valid. This note is an activityoriented companion to the study of real analysis. It is intended as a pedagogical companion for the beginner. This is a basic introduction to the idea of a metric space. If youve had a good real analysis course, then a lot though not all of the proofs below should. This book offers a unique approach to the subject which gives readers the advantage of a new perspective on ideas familiar from the analysis of a real line.
Metric space download ebook pdf, epub, tuebl, mobi. Introductory analysis i fall 2014 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. It is forwardlooking in the sense that it first and foremost aims to provide students with the concepts and techniques they need in order to follow more advanced courses in mathematical analysis and neighboring fields. Some particular properties of realvalued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability real analysis is distinguished from. The lecture notes were taken by a student in the class. Click download or read online button to get metric space book now. A metric space can be thought of as a very basic space having a geometry, with only a few axioms. Introduction when we consider properties of a reasonable function, probably the.
However, note that while metric spaces play an important role in real analysis, the study of metric spaces is by no means the same thing as real analysis. Xthe number dx,y gives us the distance between them. The particular distance function must satisfy the following conditions. For the purposes of boundedness it does not matter. In mathematics, real analysis is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real functions. Upper and lower limits of sequences of real numbers, continuous functions, differentiation, riemannstieltjes integration, unifom convergence and applications, topological results and epilogue. Download real variables with basic metric space topology download free online book chm pdf. Later i added the metric space chapter to teach math 521 at university of wisconsinmadison uw. A metric space y is clocally linearly connected if there exits c.
Some of the main results in real analysis are i cauchy sequences converge, ii for continuous functions flim n. Introduction to real analysis fall 2014 lecture notes. This book is a step towards the preparation for the study of more advanced topics in analysis such as topology. The metric structure in a normed space is very special and possesses many properties that general metric spaces do not necessarily have. A prerequisite for the course is an introductory course in real analysis. Real analysismetric spaces wikibooks, open books for an. After that he went to imsc chennai to do research in number theory. Lecture notes assignments download course materials. I introduce the idea of a metric and a metric space framed within the context of rn. A metric space is a set x where we have a notion of distance.
Sumit kumar, was an integrated phd student at iisc bangalore. Volume ii was added to teach math 41434153 at oklahoma state university osu. This book and its companion volume, advanced real analysis, systematically develop concepts. I have found that the typical beginning real analysis student simply cannot do an. Real variables with basic metric space topology download book. Metric spaces are also a kind of a bridge between real analysis and general topology. Therefore its focus lies on those parts of the theory of metric spaces which are mainly used in functional analysis. We do not develop their theory in detail, and we leave the veri. Professor copsons book, which is based on lectures given to thirdyear undergraduates at the university of st andrews, provides a more leisurely treatment of metric spaces than. This is a text for a first course in real variables for students of engineering, physics, and economics, who need to know real analysis in order to cope with the professional literature in their fields. This book on metric spaces was written by authors whose main field is analysis. It is also sometimes called a distance function or simply a distance often d is omitted and one just writes x for a metric space if it is clear from the context what metric is being used we already know a few examples of metric spaces. Some of this material is contained in optional sections of the book, but i will assume none of that and start from scratch.
The book is designed to fill the gaps left in the development of calculus as it is usually. First course in metric spaces presents a systematic and rigorous treatment of the subject of metric spaces which are mathematical objects equipped with the notion of distance. Altogether this is an interesting book for those who will continue their studies in analysis. Introduction let x be an arbitrary set, which could consist of vectors in rn, functions, sequences, matrices, etc. Let us take a look at some examples of metric spaces.
This site is like a library, use search box in the widget to get ebook that you want. A brief guide to metrics, norms, and inner products. Real variables with basic metric space topology download. Spaces is a modern introduction to real analysis at the advanced undergraduate level. For all of the lecture notes, including a table of contents, download the following file pdf. This book offers a unique approach to the subject which gives readers the advantage of a new perspective familiar from the analysis of. Find materials for this course in the pages linked along the left. Introduction to real analysis fall 2014 lecture notes vern i. If the banach space has complex scalars, then we take continuous linear function from the banach. A metric space is a set in which we can talk of the distance between. 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 most familiar is the real numbers with the usual absolute value. Lecture notes analysis ii mathematics mit opencourseware. 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.
Mathematical proof or they may be 2place predicate symbols. The abstract concepts of metric ces are often perceived as difficult. The book contains approximately 400 exercises of varying difficulty. The book treats material concerning metric spaces that is crucial for any ad. Real analysis on metric spaces columbia university.
Metric space topology, as the generalization to abstract spaces of the theory of sets of points on a line or in a plane, unifies many branches of classical analysis and is necessary introduction to functional analysis. Upper and lower limits of sequences of real numbers, continuous functions, differentiation, riemannstieltjes integration, unifom convergence. Variety of examples along with real life applications have been. Analysis on metric spaces summer school, lake arrowhead. A metric space is a pair x, d, where x is a set and d is a metric on x. Real analysis, metric spaces, function spaces, normed spaces. Real analysis on metric spaces mark dean lecture notes for fall 2014 phd class brown university 1lecture1 the.
With every metric space there is associated a topology that precisely captures the notion of continuity for the given metric. Variety of examples along with real life applications have been provided to understand. A subset of the real numbers is bounded whenever all its elements are at most some fixed distance from 0. Mathematical analysis is primarily concerned with limit processes. Let xbe a real vector space and let kkbe a norm on x. This book has been judged to meet the evaluation criteria set by. When dealing with an arbitrary metric space there may not be some natural fixed point 0. Chapter 1 metric spaces islamic university of gaza.
The abstract concepts of metric spaces are often perceived as difficult. No knowledge of metric spaces is assumed anywhere in the. Moreover the concepts of metric subspace, metric superspace. The metric topology on a metric space m is the coarsest topology on m relative to which the metric d is a continuous map from the product of m with itself to the nonnegative real numbers. Real variables with basic metric space topology dover. Pdf this chapter will introduce the reader to the concept of metrics a class of functions which. Real variables with basic metric space topology dover books on mathematics dover edition. A metric space is, essentially, a set of points together with a rule for saying how far apart two such points are. This book was published previously by pearson education. Rather than passing quickly from the definition of a metric to the more abstract concepts of convergence and continuity, the author takes the concrete notion of. The term real analysis is a little bit of a misnomer.
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