Functional analysis and infinite-dimensional geometry

Cover
Title page
Preface
1 Basic Concepts in Banach Spaces
	Hölder and Minkowski inequalities, classical spaces C[0,1], l_p, C₀, L_p[0,1]
	Operators, quotient spaces, finite-dimensional spaces, Riesz\'s lemma, separability
	Hilbert spaces, orthonormal bases, l₂
	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-Smulian, Kaplansky theorems, and Banach-Stone theorem
	Exercises
5 Structure of Banach Spaces
	Projections and complementability, Auerbach bases
	Separable spaces as subspaces of C[0,1] and quotients of l₁, Sobczyk\'s theorem, Schur\'s property of l₁
	Exercises
6 Schauder Bases
	Shrinking and boundedly complete bases, reflexivity, Mazur\'s basic sequence theorem, small perturbation lemma
	Block basis sequences, Pelczynski\'s decomposition method and subspaces of l_p, Pitt\'s theorem, Khintchine\'s inequality and subspaces of L_p 1
	Unconditional bases, James\'s theorem on containment of l₁ and c₀, James\'s space J, Bessaga-Pelczynski theorem
	Markushevich bases in separable spaces, their extension property, Johnson\'s and Plichko\'s result on l_∞
	Exercises
7 Compact Operators on Banach Spaces
	Compact and finite-rank operators, Fredholm operators, Fredholm alternative
	Eigenvalues, eigenspaces, spectrum, spectral decomposition
	Spectral theory of compact self-adjoint and compact normal operators
	Banach\'s contraction principle, nonexpansive mappings, Ryll-Nardzewski theorem, Brouwer\'s and Schauder\'s theorems, invariant subspaces
	Exercises
8 Differentiability of Norms
	Smulian\'s dual test, Kadec\'s Fréchet-smooth renorming of spaces with separable dual, Fréchet differentiability of convex functions
	More on extremal structure, Lindenstrauss\'s result on strongly exposed points and norm-attaining operators
	Exercises
9 Uniform Convexity
	Uniform convexity and uniform smoothness, l_p spaces
	Finite representability, local reflexivity, superreflexive spaces and Enflo\'s renorming, Kadec\'s and Gurarii Gurarii-James theorems
	Exercises
10 Smoothness and Structure
	Smooth and compact variational principles, subdifferential, Stegall\'s variational principle
	Partitions of unity, smooth approximation
	Lipschitz homeomorphisms, Aharoni\'s embeddings into c₀, Heinrich-Mankiewicz results on linearization of Lipschitz maps
	Homeomorphisms, Mazur\'s theorem on l_p, Kadec\'s theorem
	Smoothness in l_p and Hilbert spaces
	Countable James boundary and saturation by c₀
	Exercises
11 Weakly Compactly Generated Spaces
	Projectional resolutions, injections into c₀(Γ), Eberlein compacts, embedding into a reflexive space, locally uniformly rotund and smooth renormings
	Weakly compact operators, Davis-Figiel-Johnson-Pelczynski 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,(B_X,w) Polish
	Exercises
References
Index