Baire theorem real analysis book pdf

Analysis, real and complex analysis, and functional analysis, whose widespread use is illustrated by the fact that they have been translated into a total of languages. We have the following classical theorem of banach and mazurkiewicz which. It has the results on locally compact hausdor spaces pdf download problems and solutions in real and complex analysis as a lim sup of a sequence of measurable functions, gis measurable rudin 8, theorem i bought rudins real and complex analysis at the age of 21 at the recommendation of a fellow student, some years my senior, who not only knew much more. We shall be concerned with applications related to realvalued functions. This final text in the zakon series on mathematics analysis follows the release of the authors basic concepts of mathematics and the awardwinning mathematical analysis i and completes the material on real analysis that is the foundation for later courses in functional analysis, harmonic analysis, probability theory, etc. We require two such principles stemming from baires theorem see.

Classic applications of baire category theorem mathoverflow. Just how many points of continuity do we expect to find in the limit function. Baires theorem if x is a complete metric space, the intersection. Background reading in general topology, functional analysis, and integration theory. This is a special case of baires theorem and is given as the last problem on chapter 2 of rudins pma.

We then discuss the real numbers from both the axiomatic and constructive point of view. Understanding analysis is so wellwritten and the development of the theory so wellmotivated that exposing. The baire category theorem gives sufficient conditions for a topological space to be a baire space. The precision and clarity of presentation make the book an ideal textbook for a graduate course in real analysis while the wealth of topics treated also make the book a valuable reference work for mathematicians.

The fundamental theorem of calculus is often claimed as the central theorem of elementary calculus. Scenes from the history of real functions, baire was interested in the. See also sokals a really simple elementary proof of the uniform boundedness theorem for a proof without baire. Computable versions of baires category theorem request pdf.

It is an important tool in topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a baire space a topological space such that the intersection of countably many dense open sets is still dense. They are the uniform boundedness principle a pointwise bounded family of bounded linear operators on a banach space is bounded, the open mapping theorem a surjective bounded linear operator between. Real analysis via sequences and series springerlink. S and t have the same cardinality s t if there exists a bijection f. The baire category theorem 3 references 1richard f bass. Baire category theorem an overview sciencedirect topics. A set awhich can be written as a countable union of nowhere dense sets is called rst category, or meager.

The first 3 points of the problem are pretty easy, and i solved them. First, it is a result about holomorphic functions, so it belongs outside theusualsuspects of real analysis, topology or functional analysis. Functional analysis can best be characterized as in nite dimensional linear algebra. Mastery of the basic concepts in this book should make the analysis in such areas as complex variables, di. The baire category theorem bct is an important result in general topology and functional analysis. However, it turns out that the baire category theorem for the real line was actually proved two years earlier.

This text is evolved from authors lecture notes on the subject, and thus is very much oriented towards a pedagogical perspective. Some of the important ones are discussed in the article ibaire spacesi by j. Real analysis and multivariable calculus igor yanovsky, 2005 5 1 countability the number of elements in s is the cardinality of s. In some sense, real analysis is a pearl formed around the grain of sand provided by paradoxical sets. The only aspects of this manyheaded theorem that are used in this book. Airy function airys equation baire s theorem bolzanoweierstrass theorem cartesian product cauchy condensation test dirichlets test kummerjensen test riemann integral sequences infinite series integral test limits of functions real analysis text adoption sequence convergence. This version of elementary real analysis, second edition, is a hypertexted pdf. The book is also very helpful to graduate students in statistics and electrical engineering, two disciplines that apply measure theory. More generally, every topological space that is homeomorphic to an open subset of a complete pseudometric space is a baire space.

The goal is to produce a coherent account in a manageable scope. Mathematical analysis ii real analysis for postgraduates. I have multiple reasons to single out one particular application of baire category theorem. If that is the case at your university, you will have to supplement this book with texts in those subjects. The third is the hanhbanach extension theorem, in which completeness plays no role. It closes with a proof of the baire category theorem. New features include a discussion of infinite products, and expanded sections on metric spaces, the baire category theorem, multivariable functions, and the gamma function. Real analysisfundamental theorem of calculus wikibooks. The book can also serve as a foundation for an indepth study of real analysis giveninbookssuchas4,33,34,53,62,65listedinthebibliography. The second row is what is required in order for the. The uniform boundedness principle of functional analysis is a very important application of the baire category theorem. Apr 14, 2020 this is a collection of lecture notes ive used several times in the twosemester seniorgraduatelevel real analysis course at the university of louisville.

They are here for the use of anyone interested in such material. This is a collection of lecture notes ive used several times in the twosemester seniorgraduatelevel real analysis course at the university of louisville. New york university, courant institute of mathematical sciences. We will use some real analysis, complex analysis, and algebra, but functional analysis is not really an extension of any one of these. However, it turns out that the baire category theorem for the.

Real analysis is an important branch of mathematics which mainly deals with. Analysis on the real number line, such as one encounters in an introductory course at the advanced undergraduate level using, say, rudins principles of mathematical analysis as a textbook, constitutes only a preliminary to a vast and farreaching domain, the subject of real analysis properly so called. We conclude with a result on a characterization of baire one functions in terms of the notion of rst return recoverability, which is a topic of current research in analysis 6. Apr 21, 2017 a pointwise convergent sequence of continuous functions doesnt necessarily generate a continuous map. Your favourite application of the baire category theorem.

Introduction to real analysis university of louisville. The second row is what is required in order for the translation between one theorem and the next to be valid. Thomsonbrucknerbruckner elementary real analysis, 2nd edition 2008. For a trade paperback copy of the text, with the same numbering of theorems and. However, the baire category theorem that follows will imply that this is not the case. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext.

Any countable family of open and everywheredense sets in a given complete metric space has a nonempty, and in fact everywheredense, intersection cf. We study dierent computable versions of baires category theorem in computable analysis. We study dierent computable versions of baire s category theorem in computable analysis. Real analysislist of theorems wikibooks, open books for an. Standard references on real analysis should be consulted for more advanced topics. If ais a subset of x, then x2xis a limit point of xif each neighborhood of xcontains a point of adistinct from x. From theorem 38 and the baire category theorem follows in particular that the set of points of continuity of a function of first class from a complete metric space x to any metric space y, i.

The intersection of a sequence of dense open subsets of a complete metric space is nonempty. Baire made a number of important contributions to real analysis in addition to the. Lecture notes from the real analysis class of summer 2015 boot camp, delivered by. Similarly, as in constructive analysis, dierent logical forms of this theorem lead to dierent computational. Although it can be naturally derived when combining the formal definitions of differentiation and integration, its consequences open up a much wider field of mathematics suitable to justify the entire idea of calculus as a math discipline. Applications of the baire category theorem to real analysis.

Airy function airys equation baires theorem bolzanoweierstrass theorem cartesian product cauchy condensation test dirichlets test kummerjensen test riemann integral sequences infinite series integral test limits of functions real analysis text adoption sequence convergence. It will usually be either the name of the theorem, its immediate use for the theorem, or nonexistent. Baire made a number of important contributions to real analysis in addition to the category theorem. History baire one functions are named in honor of ren elouis baire 18741932, afrench. Baires theorem is a result on complete metric spaces which will be used in this chapter to prove some very important results on banach spaces. This restricted form applies in particular to the real line, the baire space, the. This is a preliminary version of the book an epsilon of room, i. Chapter 2 is devoted to the three fundamental principles of functional analysis. The first row is devoted to giving you, the reader, some background information for the theorem in question. However, the baire category theorem is used as a method of proving existence 1. This book is an introduction to real analysis structures.