A First Course in Functional Analysis

Metric Space: 1.1 Definitions and examples; 1.2 Inequalities of Holder and Minkowski; 1.3 Examples continued; $1_p$ spaces; 1.4 Examples continued; Function spaces; 1.5 Convergence and related notions; 1.6 Separable space, examples; 1.7 Complete space, examples; 1.8 Contractions, applications to differential and integral equations; 1.9 Completion; 1.10 Category, nowhere differentiable continuous functions; 1.11 Compactness, continuity; 1.12 Equicontinuity, application to differential equations; 1.13 Stone-Weierstrass theorems; 1.14 Normal families; 1.15 Semi-continuity, application to arc length; 1.16 Space of compact, convex sets; Exercises Banach Spaces: 2.1 Vector space; 2.2 Subspace; 2.3 Quotient space; 2.4 Dimension, Hamel basis; 2.5 Algebraic dual, second dual; 2.6 Convex sets; 2.7 Ordered groups; 2.8 Hahn-Banach theorem, separation form; 2.9 Hahn-Banach theorem, extension form; 2.10 Applications, Banach limits, invariant measure; 2.11 Banach space, dual space; 2.12 Hahn-Banach theorem in normed space; 2.13 Uniform boundedness principle, applications; 2.14 Lemma of F. Riesz, applications; 2.15 Application to compact transformations; 2.16 Applications, weak convergence, summability methods, approximate integration; 2.17 Second dual space; 2.18 Dual of $1_p$; 2.19 Dual of $C[a, b]$, Riesz representation theorem; 2.20 Open mapping and closed graph theorems; 2.21 Application, projections; 2.22 Application, Schauder expansion; 2.23 A theorem on operators in $C[0, 1]$; Exercises Measure and Integration, $L_p$ Spaces: 3.1 Lebesgue measure for bounded sets in $E_n$; 3.2 Lebesgue measure for unbounded sets; 3.3 Totally $\sigma$ finite measures; 3.4 Measurable functions, Egoroff theorem; 3.5 Convergence in measure; 3.6 Summable functions; 3.7 Fatou and Lebesgue dominated convergence theorems; 3.8 Integral as a set function; 3.9 Signed measure, decomposition into measures; 3.10 Absolute continuity and singularity of measures; 3.11 The $L_p$ spaces, completeness; 3.12 Approximation and smoothing operations; 3.13 The dual of $L_p, p>1$; 3.14 The dual of $L_1$; 3.15 The individual ergodic theorem; 3.16 $L_p$ convergence of Fourier series; 3.17 Functions whose Fourier series diverge almost everywhere; 3.18 Continuous functions which differ from all those having a given modulus; Exercises Hilbert Space: 4.1 Inner product, Hilbert space; 4.2 Basic lemma, projection theorem, dual; 4.3 Application, mean ergodic theorem; 4.4 Orthonormal sets, Fourier expansion; 4.5 Application, isoperimetric theorem; 4.6 Muntz theorem; 4.7 Dimension, Riesz-Fischer theorem; 4.8 Reproducing kernel; 4.9 Application, Bergman kernel; 4.10 Examples of complete orthonormal sets; 4.11 Systems of Haar, Rademacher, Walsh; applications; Exercises Topological Vector Spaces: 5.1 Topology; 5.2 Tychonoff theorem, application in Banach space; 5.3 Topological vector space; 5.4 Normable space; 5.5 Space of measurable functions; 5.6 Locally convex space; 5.7 Metrizable space, space of entire functions; 5.8 FK spaces; 5.9 Application to summability methods; 5.10 Ordered vector spaces; 5.11 Banach lattice; 5.12 Kothe spaces; Exercises Banach Algebras: 6.1 Definition and examples; 6.2 Adjunction of identity; 6.3 Haar measure; 6.4 Commutative Banach algebras, maximal ideals; 6.5 The set $C(\scr{M})$; 6.6 Gelfand representation for algebras with identity; 6.7 Analytic functions; 6.8 Isomorphism theorem for algebras with identity; 6.9 Algebras without identity; 6.10 Application to $L_1(G)$; Exercises References Index.