First Course in Functional Analysis.

Metric Space: 1. 1 Definitions and examples 1. 2 Inequalities of Holder and Minkowski 1. 3 Examples continued 1ₚ 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 1ₚ 2. 19 Dual of Ca, 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 C0, 1 Exercises Measure and Integration, Lₚ Spaces: 3. 1 Lebesgue measure for bounded sets in Eₙ 3. 2 Lebesgue measure for unbounded sets 3. 3 Totally 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 Lₚ spaces, completeness 3. 12 Approximation and smoothing operations 3. 13 The dual of Lₚ, p>1 3. 14 The dual of L₁ 3. 15 The individual ergodic theorem 3. 16 Lₚ 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 (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 L₁ (G) Exercises References Index.