Advanced Functional Analysis

Cover -- Half Title -- Title Page -- Copyright Page -- Contents -- Preface -- Authors -- Symbol Description -- 1. Fundamentals of Linear and Topological Spaces -- 1.1 Introduction -- 1.2 Partially ordered sets and Zorn's lemma -- 1.3 Important notations -- 1.4 Inequalities -- 1.5 Linear spaces, algebraic bases and dimension -- 1.6 Linear maps -- 1.7 Semimetric and metric spaces -- 1.8 Seminormed and normed spaces -- 1.9 Topological spaces -- 1.10 Net convergence -- 1.11 Subnets -- 1.12 Compact sets -- 2. Linear Topological Spaces -- 2.1 Introduction -- 2.2 Set arithmetic and convexity -- 2.3 Convex and affine sets -- 2.4 Balloons and cones -- 2.5 Quotient spaces -- 2.6 Supremum and weak topologies -- 2.7 Product topology -- 2.8 Properties of linear topological spaces -- 2.9 Closed maps -- 2.10 Baire's category theorem -- 2.11 Locally convex spaces -- 2.12 Quotient topologies -- 3. Linear Metric Spaces -- 3.1 Introduction -- 3.2 Paranormed spaces -- 3.3 Properties of paranormed and seminormed spaces -- 3.4 Schauder basis -- 3.5 Open mapping theorem -- 3.6 The closed graph theorem -- 3.7 Uniform boundedness principle -- 3.8 Properties of seminorms -- 3.9 The Minkowski functional -- 3.10 Local convexity -- 3.11 Metrizability -- 4. Banach Spaces -- 4.1 Introduction -- 4.2 Some basic properties -- 4.3 Examples -- 4.4 Bounded linear operators -- 4.5 The Hahn-Banach theorem -- 4.6 Important theorems -- 4.7 Representation theorems -- 4.8 Reflexivity -- 4.9 Adjoint operators -- 4.10 Quotient spaces -- 4.11 Cauchy nets, summable families -- 4.12 Equivalent norms -- 4.13 Compactness and the Riesz lemma -- 4.14 Compact operators -- 4.15 Operators with closed range -- 5. Hilbert Spaces -- 5.1 Introduction -- 5.2 Inner product spaces -- 5.3 Elementary properties and Hilbert spaces -- 5.4 Examples -- 5.5 Theorem P. Jordan-J. von Neumann -- 5.6 Orthogonality.