Book:Marián Fabian/Functional Analysis and Infinite-Dimensional Geometry

Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos Santalucía, Jan Pelant and Václav Zizler: Functional Analysis and Infinite-Dimensional Geometry

Published $\text {2001}$

Subject Matter

• Functional Analysis

Contents

Preface

1 Basic Concepts in Banach Spaces

Hölder and Minkowski inequalities, classical spaces $C \closedint 0 1$, $\ell_p$, $c_0$, $L_p \closedint 0 1$

Operators, quotient spaces, finite-dimensional spaces, Riesz's lemma, separability

Hilbert spaces, orthonormal bases, $\ell_2$

Exercises

2 Hahn-Banach and Banach Open Mapping Theorems

Hahn-Banach extension and separation theorems

Duals of classical spaces

Banach open mapping theorem, closed graph theorem, dual operators

Exercises

3 Weak Topologies

Weak and weak star topology, Banach-Steinhaus uniform boundedness principle, Alaoglu's and Goldstine's theorem, reflexivity

Extreme points, Krein-Milman theorem, James boundary, Ekeland's variational principle, Bishop-Phelps theorem

Exercises

4 Locally Convex Spaces

Local bases, bounded sets, metrizability and normability, finite-dimensional spaces, distributions

Bipolar theorem, Mackey topology

Carathéodory and Choquet representation; Banach-Dieudonné, Eberlein-Šmulian theorem, Kaplansky theorems, and Banach-Stone theorem

Exercises

5 Structure of Banach Spaces

Projections and complementability, Auerbach bases

Separable spaces as subspaces of $C \closedint 0 1$ and quotients of $\ell_1$, Sobczyk's theorem, Schur's property of $\ell_1$

Exercises

6 Schauder Bases

Shrinking and boundedly complete bases, reflexivity, Mazur’s basic sequence theorem, small perturbation lemma

Bases in classical spaces: block basis sequences, Pełczyński decomposition method and subspaces of $\ell_p$, Pitt’s theorem, Khintchine’s inequality and subspaces of $L_p$

Unconditional bases, James’s theorem on containment of $\ell_1$ and $c_0$, James's space $J$, Bessaga-Pełczyński theorem

Markushevich bases: existence for separable spaces, extension property, Johnson’s and Plichko’s result on $\ell_\infty$

Exercises

7 Compact Operators on Banach Spaces

Compact operators and finite rank operators, Fredholm operators, Fredholm alternative

Spectral theory: eigenvalues, spectrum, resolvent, eigenspaces

Self-adjoint operators, spectral theory of compact self-adjoint and compact normal operators

Fixed points: Banach’s contraction principle, non-expansive mappings, Ryll-Nardzewski theorem, Brouwer’s and Schauder’s theorems, invariant subspaces

Exercises

8 Differentiability of Norms

Šmulian’s dual test, Kadec’s Fréchet-smooth renorming of spaces with separable dual, Fréchet differentiability of convex functions

Extremal structure, Lindenstrauss’ result on strongly exposed points and norm attaining operators

Exercises

9 Uniform Convexity

Uniform convexity and uniform smoothness, $\ell_p$ spaces

Finite representability, local reflexivity, superreflexive spaces and Enflo's renorming, Kadec’s and Gurarii-Gurarii-James theorems

Exercises

10 Smoothness and Structure

Variational principles (smooth and compact), subdifferential, Stegall’s variational principle

Smooth approximation: partitions of unity

Lipschitz homeomorphisms, Aharoni’s embeddings into $c_0$, Heinrich-Mankiewicz results on linearization of Lipschitz maps

Homeomorphisms: Mazur’s theorem on $\ell_p$, Kadec’s theorem

Smoothness in $\ell_p$, Hilbert spaces

Countable James boundary and saturation by $c_0$

Exercises

11 Weakly Compactly Generated Spaces

Projectional resolutions, injections into $\map {c_0} \Gamma$, Eberlein compacts, embedding into a reflexive space, locally uniformly rotund and smooth renormings

Weakly compact operators, Davis-Figiel-Johnson-Pełczyński factorization, absolutely summing operators, Pietsch factorization, Dunford-Pettis property

Quasicomplements

Exercises

12 Topics in Weak Topology

Eberlein compacts, metrizable subspaces

Uniform Eberlein compacts, scattered compacts

Weakly Lindelöf spaces, property C

Corson compacts, weak pseudocompactness in Banach spaces, $\struct {B_X, w}$ Polish

Exercises

References

Index