# Real Analysis: Modern Techniques and Their Applications

## Gerald B. Folland

Language: English

Pages: 416

ISBN: 0471317160

Format: PDF / Kindle (mobi) / ePub

An in-depth look at real analysis and its applications-now expanded and revised.

This new edition of the widely used analysis book continues to cover real analysis in greater detail and at a more advanced level than most books on the subject. Encompassing several subjects that underlie much of modern analysis, the book focuses on measure and integration theory, point set topology, and the basics of functional analysis. It illustrates the use of the general theories and introduces readers to other branches of analysis such as Fourier analysis, distribution theory, and probability theory.

This edition is bolstered in content as well as in scope-extending its usefulness to students outside of pure analysis as well as those interested in dynamical systems. The numerous exercises, extensive bibliography, and review chapter on sets and metric spaces make Real Analysis: Modern Techniques and Their Applications, Second Edition invaluable for students in graduate-level analysis courses. New features include:
* Revised material on the n-dimensional Lebesgue integral.
* An improved proof of Tychonoff's theorem.
* Expanded material on Fourier analysis.
* A newly written chapter devoted to distributions and differential equations.
* Updated material on Hausdorff dimension and fractal dimension.

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9.3. By Proposition 5.15, the continuity of F implies that there exist m, N, C such that and hence by (8.12), The same reasoning applies with replaced by is slowly increasing. Next, by Proposition 9.3 we know that the equation holds when ϕ, . By Proposition 9.9, if we can find sequences {ϕj} and {j} in Cc∞ that converge to ϕ and in . Then in by (the proof of) Proposition 8.11, so . On the other hand, the preceding estimates show that |F * j(x)| ≤ C(1 + |x|)m with C and m independent of j, and

�1.5. That is, where See Figure 11.1a. Fig. 11.1 The first three approximations to (a) the Cantor set C3/s, (b) the Sierpiński gasket, and (c) the snowflake curve. The Sierpiński gasket Γ is the subset of 2 obtained from a solid triangle by dividing it into four equal subtriangles by bisecting the sides, deleting the middle subtriangle, and then iterating. Thus, if we take the initial triangle Δ to be the closed triangular region with vertices (0,0), (1,0), and (1/2, 1), then Γ = , where S =

if μ(X) = ∞. 39. If fn → f almost uniformly, then fn → f a.e. and in measure. 40. In Egoroff’s theorem, the hypothesis “μ(X) < ∞” can be replaced by “|fn| ≤ g for all n, where g L1 (μ).” 41. If μ is σ-finite and fn → f a.e., there exist measurable E1, E2,… ⊂ X such that μ((∪∞1 Ej)c) = 0 and fn → f uniformly on each Ej. 42. Let μ be counting measure on . Then fn → f in measure iff fn → f uniformly. 43. Suppose that μ(X) < ∞ and f : X × [0, 1] → is a function such that f(·, y) is measurable

functional F on such that F(x) ≥ 0 for x P and F| = f. (Consider p(x) = inf {f(y) : y and x ≤ y}.) 5.3 THE BAIRE CATEGORY THEOREM AND ITS CONSEQUENCES In this section we present an important theorem about complete metric spaces and use it to obtain some fundamental results concerning linear maps between Banach spaces. 5.9 The Baire Category Theorem. Let X be a complete metric space. a. If {Un}∞1 is a sequence of open dense subsets of X, then ∩∞1 Un is dense in X. b. X is not a countable

on Q; the point of view will be clear from the context when it matters. Exercises 1. Prove the product rule for partial derivatives as stated in the text. Deduce that for some constants cγδ and c’γδ with cγδ = c’γδ = 0 unless |γ| < |α| and |δ| < |β|. 2. Observe that the binomial theorem can be written as follows: Prove the following generalizations: a. The multinomial theorem: If x n, b. The n-dimensional binomial theorem: If x, y n, 3. Let η(t) = e−1/t for t > 0, η(t) = 0 for t < 0. a.