Baire theorem real analysis book

New topics include material on the irrationality of pi, the baire category theorem, newtons method and the secant method, and continuous nowheredifferentiable functions. C, where c contains all the real numbers of the interval 0,1. The baire category theorem bct is an important result in general topology and functional analysis. Bct2 every locally compact hausdorff space is a baire space. 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. The baire category theorem is an illuminating example of a theorem that can be used to demonstrate that one classical theorem can have several different computational interpretations. The standard textbook proof of the uniform boundedness theorem is the one i. Thus every completely metrizable topological space is a baire space. In addition, several topics from real analysis and elementary topology which are not usually covered in undergraduate courses totally disconnected spaces and cantor sets, points of discontinuity of maps and the baire category theorem, surjectivity of isometries of compact metric spaces are also included in the book.

Integers and rational numbers, building the real numbers, series, topological concepts, functions, limits, and continuity, cardinality, representations of the real numbers, the derivative and the riemann integral, vector and function spaces, finite taylormaclaurin expansions, integrals on rectangles. The second row is what is required in order for the translation between one theorem and the next to be valid. We shall be concerned with applications related to realvalued functions. We require two such principles stemming from baires theorem see. The baire category theorem 3 references 1richard f bass. Background reading in general topology, functional analysis, and integration theory. The material covered includes classic topics from both set theory and real analysis courses, such as countable and uncountable sets, countable ordinals, the continuum problem, the cantorschroderbernstein theorem, continuous functions, uniform convergence, zorns lemma, borel sets, baire functions, lebesgue measure, and riemann integrable.

If each has no interior point, also has no interior point. Strange functions in real analysis, third edition differs from the previous editions in that it includes five new chapters as well as two appendices. An important new graduate text that motivates the reader by providing the historical evolution of modern analysis. Computable versions of baires category theorem request pdf. The book is intended as an introduction to this kind of analysis. It will usually be either the name of the theorem, its immediate use for the theorem, or nonexistent. Real analysisfundamental theorem of calculus wikibooks. This is a collection of lecture notes ive used several times in the twosemester seniorgraduatelevel real analysis course at the university of louisville. Strange functions in real analysis 3rd edition routledge book. Baire made a number of important contributions to real analysis in addition to the. Applications of the baire category theorem to real analysis. In real analysis we need to deal with possibly wild functions on r and fairly general subsets of r, and as a result a rm grounding in basic set theory is helpful.

The first 3 points of the problem are pretty easy, and i solved them. This chapter focuses on such structure theorems for several familiar metric spaces that arise in the theory of realvalued functions. Real analysis 3rd ed speaking tiger books speaking. According to the baireosgood theorem, the limit function is mostly continuous. However, it turns out that the baire category theorem for the real line was actually proved two years earlier. Baire made a number of important contributions to real analysis in addition to the category theorem. The baire category theorem oregon state university.

But this requires that the book stand out in some way or another. I have taught the beginning graduate course in real variables and functional analysis three times in the last. The second edition preserves the books clear and concise style, illuminating discussions, and simple, wellmotivated proofs. I have multiple reasons to single out one particular application of baire category theorem. Real analysislist of theorems wikibooks, open books for an. The fundamental theorem of calculus is often claimed as the central theorem of elementary calculus. More generally, every topological space which is homeomorphic to an open subset of a complete pseudometric space is a baire space. Baires theorem if x is a complete metric space, the intersection.

Thanks for contributing an answer to mathematics stack exchange. Real analysis via sequences and series springerlink. The course assumes that the student has seen the basics of real variable theory and point set topology. The only aspects of this manyheaded theorem that are used in this book. First, it is a result about holomorphic functions, so it belongs outside theusualsuspects of real analysis, topology or functional analysis. It could be used to supplement a standard course in real analysis, as the basis for a seminar, or for inde pendent study. We start with the careful discussion of the axiom of completeness and proceed to the study of the basic concepts of limits, continuity, riemann integrability, and differentiability. For beginning graduatelevel courses in real analysis, measure theory, lebesque integration, and functional analysis. 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.

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limitrelated structure e. This text is evolved from authors lecture notes on the subject, and thus is very much oriented towards a pedagogical perspective. 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. As lindstrom points out in his preface, real analysis textbooks can be categorized as being elementary, intermediate, or graduate level. Baire category theorem an overview sciencedirect topics. This earlier book aims to develop problem solving skills in classical analysis and offers a much larger selection of challenging problems. 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. It is primarily expository, but a few refinements of known results are included, notably theorem 15. A new book adds to the breadth of resources available to students and instructors. 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. After introducing the basic concepts, the author begins with cantor and peanotype functions, then moves to functions whose constructions require essentially noneffective methods. Baire category theorem, typical, typical continuous functions. This restricted form applies in particular to the real line, the baire space, the.

Real analysislist of theorems wikibooks, open books for. The theorem was proved by french mathematician renelouis baire in his 1899 doctoral thesis. 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. Some of the important ones are discussed in the article ibaire spacesi by j. A problem book in real analysis mathematical association of. We have the following classical theorem of banach and mazurkiewicz which. Rudin, real and complex analysis, mcgrawhill series in higher mathematics, 1974. Topical survey applications of the baire category theorem. Then every neighborhood of pintersects ac and therefore pcannot be an interior point of a. The first row is devoted to giving you, the reader, some background information for the theorem in question. The baire category theorem bjg may 2009 the theorem. Sep 27, 2007 this is a special case of baire s theorem and is given as the last problem on chapter 2 of rudins pma.

The elements of the topology of metrics spaces are presented. At the heart of these theorems, we often find the baire category theorem as a. For a more challenging collection, see problems in real analysis. The course is the rigorous introduction to real analysis. The book is also very helpful to graduate students in statistics and electrical engineering, two disciplines that apply measure theory. Baires theorem and its applications the completeness of. 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. For a trade paperback copy of the text, with the same numbering of theorems and exercises but with di. They are here for the use of anyone interested in such material. Any countable family of open and everywheredense sets in a given complete metric space has a nonempty, and in fact everywheredense, intersection cf.

Measure and category a survey of the analogies between. The equivalence of different forms of baires category. It can, however, be used to obtain some rather interesting results. Im reading a chinese text book real analysis by zhou minqiang, one of its conclusion is baire theorem.

More importantly, the entire text has been revised and contains more detailed explanations of the presented material. This version of elementary real analysis, second edition, is a hypertexted pdf. Rudins hint tells me imitate the proof of theorem 2. The third is the hanhbanach extension theorem, in which completeness plays no role.