Functional Analysis and Infinite-Dimensional Geometry

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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