Functional Analysis and Infinite-Dimensional Geometry

Frontmatter

1. Basic Concepts in Banach Spaces

Abstract

Most of the theory presented in this text is valid for both real and complex scalar fields. When the proofs are similar, we formulate the theorems without specifying the field over which we are working. When theorems are not valid in both fields or their proofs are different, we specify the scalar field in the formulation of a theorem. K denotes simultaneously the real (R) or complex (C) scalar field. We use N for {1,2,...}. All topologies are assumed to be Hausdorff. In particular, by a compact space we mean a compact Hausdorff space.

Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos Santalucía, Jan Pelant, Václav Zizler

2. Hahn—Banach and Banach Open Mapping Theorems

Abstract

A real-valued function p on a vector space X is called a positively homogeneous sublinear functional if for all x, y ϵ X and α ≥ 0 it satisfies

$$p(\alpha x) = \alpha p(x)\;\;and\;\;p(x + y) \leqslant p(x) + p(y)$$

.

Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos Santalucía, Jan Pelant, Václav Zizler

3. Weak Topologies

Abstract

Given a normed space (X, ‖ · ‖), by X** we denote the space (X*)* with the norm \( \left\| F \right\| = \mathop {\sup }\limits_{f \in {B_x}*} \left| {F(f)} \right| \). We define higher duals by induction as X*** = (X**)*, etc.

Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos Santalucía, Jan Pelant, Václav Zizler

4. Locally Convex Spaces

Abstract

In this chapter, we restrict ourselves to the real scalar field.

Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos Santalucía, Jan Pelant, Václav Zizler

5. Structure of Banach Spaces

Abstract

Let X be a vector space. A linear map P:X → X is called a projection onto a subspace Y of X if P(X) = Y and P(y) = y for every y ∈ Y.

Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos Santalucía, Jan Pelant, Václav Zizler

6. Schauder Bases

Abstract

Let X be an infinite-dimensional normed linear space. A sequence \(\left\{ {{e_i}} \right\}_i^\infty = 1\) in X is called a Schauder basis of X if for every x ∈ X there is a unique sequence of scalars \(\left( {{a_i}} \right)_i^\infty = 1\) called the coordinates of x, such that \(x = \sum\limits_{i = 1}^\infty {{a_i}{e_i}}\).

Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos Santalucía, Jan Pelant, Václav Zizler

7. Compact Operators on Banach Spaces

Abstract

Let X and Y be Banach spaces.

Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos Santalucía, Jan Pelant, Václav Zizler

8. Differentiability of Norms

Abstract

Let f be a real-valued function on an open subset U of a Banach space X. Let x ∈ U. We say that f is Gâteaux differentiable at x if there is F ∈ X* such that

$$\mathop {\lim }\limits_{t \to 0} \frac{{f(x + th) - f(x)}}{t} = F(h)$$

for every h ∈ X.

Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos Santalucía, Jan Pelant, Václav Zizler

9. Uniform Convexity

Abstract

Let (X, ‖ · ‖) be a Banach space. For every ε ∈ (0, 2], we define the modulus of convexity (or rotundity) of ‖ · ‖ by

$${\delta _X}\left( \varepsilon \right) = \inf \left\{ {1 - \left\| {\frac{{x + y}}{2}} \right\|;\,x,y \in {B_X},\,\left\| {x - y} \right\| \geqslant \varepsilon } \right\}$$

.

Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos Santalucía, Jan Pelant, Václav Zizler

10. Smoothness and Structure

Abstract

Let X be a Banach space, n э N.

Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos Santalucía, Jan Pelant, Václav Zizler

11. Weakly Compactly Generated Spaces

Abstract

A Banach space X is called weakly compactly generated (WCG) if there is a weakly compact set K in X such that X = span̄ (K).

Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos Santalucía, Jan Pelant, Václav Zizler

12. Topics in Weak Topology

Abstract

Let K be a compact set, and let L be a pointwise closed set in C (K).

Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos Santalucía, Jan Pelant, Václav Zizler

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