# Real Mathematical Analysis (Undergraduate Texts in Mathematics)

Language: English

Pages: 478

ISBN: 3319177702

Format: PDF / Kindle (mobi) / ePub

Based on an honors course taught by the author at UC Berkeley, this introduction to undergraduate real analysis gives a different emphasis by stressing the importance of pictures and hard problems. Topics include: a natural construction of the real numbers, four-dimensional visualization, basic point-set topology, function spaces, multivariable calculus via differential forms (leading to a simple proof of the Brouwer Fixed Point Theorem), and a pictorial treatment of Lebesgue theory. Over 150 detailed illustrations elucidate abstract concepts and salient points in proofs. The exposition is informal and relaxed, with many helpful asides, examples, some jokes, and occasional comments from mathematicians, such as Littlewood, Dieudonné, and Osserman. This book thus succeeds in being more comprehensive, more comprehensible, and more enjoyable, than standard introductions to analysis.

New to the second edition of *Real Mathematical Analysis* is a presentation of Lebesgue integration done almost entirely using the undergraph approach of Burkill. Payoffs include: concise picture proofs of the Monotone and Dominated Convergence Theorems, a one-line/one-picture proof of Fubini's theorem from Cavalieri’s Principle, and, in many cases, the ability to *see* an integral result from measure theory. The presentation includes Vitali’s Covering Lemma, density points ― which are rarely treated in books at this level ― and the almost everywhere differentiability of monotone functions. Several new exercises now join a collection of over 500 exercises that pose interesting challenges and introduce special topics to the student keen on mastering this beautiful subject.

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functions, lab f(x)g(x) dx < It would be challenging to prove such an inequality from scratch, would it not? A norm on a vector space V is any function I II: V -+ JR. with the three properties of vector length: namely, if v, W E V and A E JR. then 28 Real Numbers Chapter I Ilvll :::: 0 and Ilvll = 0 if and only if v = 0, IIA-vll = IA-Illvll, IIv + wll ::: IIvll + Ilwll. An inner product ( , ) defines a nonn as II v II = ~ , but not all nonns come from inner products. The unit sphere {v E

following simple lemma about dividing a metric space M into small pieces. A piece of M is any compact, non-empty subset of M. c ---- Figure 50 (J' surjects C onto M. 100 A Taste of Topology Chapter 2 66 Piece Lemma A compact metric space M is the union of dyadically many small pieces. Specifically, given E > 0, there exist 2k pieces of M, each with diameter::::: E, whose union is M. Proof (Recall that "dyadic" refers to powers of 2.) Cover M with neighborhoods of radius E /2. By

corresponding to all words 8 of length::: n1 and M = U (U (Ma)fJ) = U lal=1I1 IfJl=n2 + n2, M o. lol=nl + n 2 For each word 8 of length n 1 + n2 is expressed uniquely as a compound word 8=a.B where lal = nl and I.BI = n2. This extends the previous filtration to ?ltn , I ::: n ::: nl + n2, and again (c) is automatic. Nothing stops us from passing to pieces of ever smaller diameter, which produces the desired dyadic filtration ?It = U?ltn • 0 Consider the intervals Ca that define the

map id : Cmax --+ ent where Cmax is the metric space C ([a, b], JR) of continuous real valued functions defined on [a, b], equipped with the max metric dmax(f, g) = max If(x) - g(x)l, and Cint is C([a, b], JR) equipped with the integral metric, dint(f, g) = lb If(x) - g(x)1 dx. Show that id is a continuous linear bijection (an isomorphism) but its inverse is not continuous. 38. Let II II be any norm on JRm and let B = {x E JRm : IIxll :::: I}. Prove that B is compact. [Hint: It suffices to

easier for a differentiable function? The idea is to reduce an abstract problem to its simplest concrete manifestation, rather like a metaphor in reverse. At the minimum, look for at least one instance in which you can solve the problem, and build from there. Moral If you do not see how to solve a problem in complete generality, first solve it in some special cases. Here is the second piece of advice. Buy a notebook. In it keep a diary of your own opinions about the mathematics you are learning.