Analysis II: Third Edition (Texts and Readings in Mathematics)
Language: English
Pages: 218
ISBN: 9380250657
Format: PDF / Kindle (mobi) / ePub
This is part two of a two-volume book on real analysis and is intended for senior undergraduate students of mathematics who have already been exposed to calculus. The emphasis is on rigour and foundations of analysis. Beginning with the construction of the number systems and set theory, the book discusses the basics of analysis (limits, series, continuity, differentiation, Riemann integration), through to power series, several variable calculus and Fourier analysis, and then finally the Lebesgue integral. These are almost entirely set in the concrete setting of the real line and Euclidean spaces, although there is some material on abstract metric and topological spaces. The book also has appendices on mathematical logic and the decimal system. The entire text (omitting some less central topics) can be taught in two quarters of 25–30 lectures each. The course material is deeply intertwined with the exercises, as it is intended that the student actively learn the material (and practice thinking and writing rigorously) by proving several of the key results in the theory.
Geometric Analysis of the Bergman Kernel and Metric (Graduate Texts in Mathematics, Volume 268)
Elementary Analysis: The Theory of Calculus (2nd Edition) (Undergraduate Texts in Mathematics)
Paradoxes from A to Z (3rd Edition)
Playing with Infinity: Mathematical Explorations and Excursions
12.4.7. D In contrast, an incomplete metric space such as (Q, d) may be considered closed in some spaces (for instance, Q is closed in Q) but not in others (for instance, Q is not closed in R). Indeed, it turns out that given any incomplete metric space (X, d), there exists a completion (X, d), which is a larger metric space containing (X, d) which is complete, and such that X is not closed in X (indeed, the closure of X in (X, d) will be all of X); see Exercise 12.4.8. For instance, one
convergence). Let (f(n))~=l be a sequence of functions from one metric space (X, dx) to another (Y, dy), and let f : X ---+ Y be another function. We say that (f(n))~ 1 converges pointwise to f on X if we have lim f(n)(x) n-+oo = f(x) for all x EX, i.e. lim dy(f(n)(x), f(x)) n-+oo = 0. Or in other words, for every x and every c > 0 there exists N > 0 such that dy(f(n)(x), f(x)) < c for every n > N. We call the function f the pointwise limit of the functions f(n). Remark 14.2.2. Note that
> 0 such that I{ f(n)- { J[a,b] fl :::; 2c-(b- a) J[a,b] for all n 2:: N. Since c-is arbitrary, we see that fra,b] j(n) converges to i[a,b] f as desired. D To rephrase Theorem 14.6.1: we can rearrange limits and integrals (on compact intervals [a, b]), lim { n-->oo J[a,b] jCn) = f lim jCnl, J[a,b] n-->oo provided that the convergence is uniform. This should be contrasted with Example 14.2.5 and Example 1.2.9. There is an analogue of this theorem for series: Corollary 14.6.2. Let [a, b]
: --+ R, g : R ~ R, and h : R ~ R be continuous, compactly supported functions. Then the following statements are true. R (a) The convolution f ported function. *g is also a continuous, compactly sup- (b) (Convolution is commutative) We have f*g = g*f; in other words f * g(x) =I: =I: f(y)g(x- y) dy g(y)f(x- y) dy = g * f(x). (c) (Convolution is linear) We have f * (g +h)= f * g + f *h. Also, for any real number c, we have f * (cg) = ( cf) * g = c(f *g). Proof. See Exercise 14.8.11. 0
(Cauchy-Schwarz inequality) We have l(f,g}l:::; llfll2llgll2(c) (Triangle inequality) We have llf + gll2:::; llfll2 + llgll2· (d) (Pythagoras' theorem) If (!,g)= 0, then llgll~- II!+ gil~= 11!11~+ (e) {Homogeneity) We have llcfll2 = lclllfll2 for all c E C. Proof. See Exercise 16.2.4. D In light of Pythagoras' theorem, we sometimes say that f and g are orthogonal iff (!,g) = 0. We can now define the L 2 metricd£2 on C(R/Z; C) by defining d£2(f,g) :=II!- gll2 = ( { J[o,l] lf(x)- g(xW dx) 112
