Real analysis, metric spaces, function spaces, normed spaces. Introduction to real analysis fall 2014 lecture notes. Metric spaces are generalizations of the real line, in which some of the theorems that hold for r remain valid. Real variables with basic metric space topology dover. The metric structure in a normed space is very special and possesses many properties that general metric spaces do not necessarily have. No knowledge of metric spaces is assumed anywhere in the. Lecture notes assignments download course materials. Find materials for this course in the pages linked along the left. Download real variables with basic metric space topology download free online book chm pdf.
If youve had a good real analysis course, then a lot though not all of the proofs below should. After that he went to imsc chennai to do research in number theory. Metric space download ebook pdf, epub, tuebl, mobi. 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. I have found that the typical beginning real analysis student simply cannot do an. Let us take a look at some examples of metric spaces.
Some of the main results in real analysis are i cauchy sequences converge, ii for continuous functions flim n. Metric spaces are also a kind of a bridge between real analysis and general topology. This book on metric spaces was written by authors whose main field is analysis. The abstract concepts of metric spaces are often perceived as difficult. The book contains approximately 400 exercises of varying difficulty. When dealing with an arbitrary metric space there may not be some natural fixed point 0. It is intended as a pedagogical companion for the beginner. Volume ii was added to teach math 41434153 at oklahoma state university osu. Click download or read online button to get metric space book now. A metric space is, essentially, a set of points together with a rule for saying how far apart two such points are.
This book has been judged to meet the evaluation criteria set by. This note is an activityoriented companion to the study of real analysis. This book offers a unique approach to the subject which gives readers the advantage of a new perspective familiar from the analysis of. The term real analysis is a little bit of a misnomer. A metric space y is clocally linearly connected if there exits c. Variety of examples along with real life applications have been provided to understand. The most familiar is the real numbers with the usual absolute value.
A metric space is a set in which we can talk of the distance between. A metric space can be thought of as a very basic space having a geometry, with only a few axioms. Therefore its focus lies on those parts of the theory of metric spaces which are mainly used in functional analysis. Let xbe a real vector space and let kkbe a norm on x. I introduce the idea of a metric and a metric space framed within the context of rn. This site is like a library, use search box in the widget to get ebook that you want. This book is a step towards the preparation for the study of more advanced topics in analysis such as topology. A metric space is a set x where we have a notion of distance. We do not develop their theory in detail, and we leave the veri. Pdf this chapter will introduce the reader to the concept of metrics a class of functions which. This is a basic introduction to the idea of a metric space. Real analysismetric spaces wikibooks, open books for an. Xthe number dx,y gives us the distance between them. 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.
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. Introduction when we consider properties of a reasonable function, probably the. 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. A metric space is a pair x, d, where x is a set and d is a metric on x. A subset of the real numbers is bounded whenever all its elements are at most some fixed distance from 0. Real variables with basic metric space topology download book. Some of this material is contained in optional sections of the book, but i will assume none of that and start from scratch. Introduction to real analysis fall 2014 lecture notes vern i. 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. Real analysis on metric spaces columbia university. For the purposes of boundedness it does not matter. Upper and lower limits of sequences of real numbers, continuous functions, differentiation, riemannstieltjes integration, unifom convergence and applications, topological results and epilogue.
The lecture notes were taken by a student in the class. 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. 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 is a lecture notes on distributions without locally convex spaces, very basic functional analysis, lp spaces, sobolev spaces, bounded operators, spectral theory for compact self adjoint operators and the fourier transform. If the banach space has complex scalars, then we take continuous linear function from the banach. Real analysis on metric spaces mark dean lecture notes for fall 2014 phd class brown university 1lecture1 the. Chapter 1 metric spaces islamic university of gaza. 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. Real variables with basic metric space topology download. 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. Spaces is a modern introduction to real analysis at the advanced undergraduate level. Later i added the metric space chapter to teach math 521 at university of wisconsinmadison uw. 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.
Sumit kumar, was an integrated phd student at iisc bangalore. Altogether this is an interesting book for those who will continue their studies in analysis. A brief guide to metrics, norms, and inner products. With every metric space there is associated a topology that precisely captures the notion of continuity for the given metric. Mathematical analysis is primarily concerned with limit processes. A prerequisite for the course is an introductory course in real analysis. 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. The particular distance function must satisfy the following conditions. Lecture notes analysis ii mathematics mit opencourseware.
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. Introduction let x be an arbitrary set, which could consist of vectors in rn, functions, sequences, matrices, etc. This book and its companion volume, advanced real analysis, systematically develop concepts. Mathematical proof or they may be 2place predicate symbols. 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. The book is designed to fill the gaps left in the development of calculus as it is usually. Analysis on metric spaces summer school, lake arrowhead. 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 book treats material concerning metric spaces that is crucial for any ad. For all of the lecture notes, including a table of contents, download the following file pdf. Moreover the concepts of metric subspace, metric superspace. Variety of examples along with real life applications have been.
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