Download Analyticity in infinite dimensional spaces by Michel Hervé PDF

By Michel Hervé

Show description

Read or Download Analyticity in infinite dimensional spaces PDF

Best mathematical analysis books

Introduction to Fourier analysis and wavelets

This booklet offers a concrete advent to a few themes in harmonic research, available on the early graduate point or, in certain cases, at an higher undergraduate point. invaluable necessities to utilizing the textual content are rudiments of the Lebesgue degree and integration at the genuine line. It starts with a radical remedy of Fourier sequence at the circle and their purposes to approximation conception, chance, and airplane geometry (the isoperimetric theorem).

Summability of Multi-Dimensional Fourier Series and Hardy Spaces

The historical past of martingale conception is going again to the early fifties while Doob [57] mentioned the relationship among martingales and analytic services. at the foundation of Burkholder's medical achievements the mar­ tingale conception can completely good be utilized in advanced research and within the thought of classical Hardy areas.

Extra info for Analyticity in infinite dimensional spaces

Sample text

3 of the previous section. 1 (The Mazur–Ulam theorem) If L : E → F is an isometry of a real normed space (E, . E ) onto a real normed space (F, . F ) with L(0) = 0, then L is a linear mapping. In order to prove this, we introduce some ideas concerning the geometry of metric spaces, of interest in their own right. First, suppose that x, y, z are elements of a metric space. We say that y is between x and z if d(x, y) + d(y, z) = d(x, z), and we say that y is halfway between x and z if d(x, y) = 324 Metric spaces and normed spaces d(y, z) = 12 d(x, z).

3 Suppose that (X, d), (Y, ρ) and (Z, σ) are metric spaces, that f is a continuous surjective mapping of (X, d) onto (Y, ρ) and that g : (Y, ρ) → (Z, σ) is continuous. Show that if g ◦ f is a homeomorphism of (X, d) onto (Z, σ) then f is a homeomorphism of (X, d) onto (Y, ρ) and g is a homeomorphism of (Y, ρ) onto (Z, σ). 4 Show that the punctured unit sphere {x ∈ Rd : x = 1} \ {(1, 0, . . , 0)} of Rd , with its usual metric, is homeomorphic to Rd−1 . 5 Give an example of three metric subspaces A, B and C of R such that A ⊂ B ⊂ C, A and C are homeomorphic, and B and C are not homeomorphic.

If x ∈ Nδ (a), then f (x) E ≤ f (x) − f (a) E + f (a) E ≤ η + M , so that λ(x)f (x) − λ(a)f (a) E = (λ(x) − λ(a))f (x) + λ(a)(f (x) − f (a)) ≤ |λ(x) − λ(a)|. f (x) E E + |λ(a)|. f (x) − f (a) E ≤ η(η + M ) + M η ≤ . (v) Suppose that > 0. Let η = |λ(a)|2 /2. There exists δ > 0 such that |λ(x) − λ(a)| < max(|λ(a)|/2, η) for x ∈ Nδ (a). If x ∈ Nδ (a), then |λ(x)| ≥ |λ(a)|/2, and so 1 2η 1 λ(a) − λ(x) − = ≤ = . 4 (The sandwich principle) Suppose that f , g and h are real-valued functions on a metric space (X, d), and that there exists η > 0 such that f (x) ≤ g(x) ≤ h(x) for all x ∈ Nη (a), and that f (a) = g(a) = h(a).

Download PDF sample

Rated 4.07 of 5 – based on 48 votes