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Functional Analysis by Theo Buhler and Dietmar A. Salamon


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Functional Analysis written by Theo Buhler and Dietmar A. Salamon . This is an other book of mathematics cover the following topics.


  • 1. Foundations
    1.1 Metric Spaces and Compact Sets
    1.1.1 Banach Spaces
    1.1.2 Compact Sets
    1.1.3 The Arzel`a–Ascoli Theorem
    1.2 Finite-Dimensional Banach Spaces
    1.2.1 Bounded Linear Operators
    1.2.2 Finite-Dimensional Normed Vector Spaces
    1.2.3 Quotient and Product Spaces
    1.3 The Dual Space
    1.3.1 The Banach Space of Bounded Linear Operators
    1.3.2 Examples of Dual Spaces
    1.3.3 Hilbert Spaces
    1.4 Banach Algebras
    1.5 The Baire Category Theorem
    1.6 Problems

  • 2. Principles of Functional Analysis
    2.1 Uniform Boundedness
    2.2 Open Mappings and Closed Graphs
    2.2.1 The Open Mapping Theorem
    2.2.2 The Closed Graph Theorem
    2.2.3 Closeable Operators
    2.3 Hahn–Banach and Convexity
    2.3.1 The Hahn–Banach Theorem
    2.3.2 Positive Linear Functionals
    2.3.3 Separation of Convex Sets
    2.3.4 The Closure of a Linear Subspace
    2.3.5 Complemented Subspaces
    2.3.6 Orthonormal Bases
    2.4 Reflexive Banach Spaces
    2.4.1 The Bidual Space
    2.4.2 Reflexive Banach Spaces
    2.4.3 Separable Banach Spaces
    2.4.4 The James Space
    2.5 Problems

  • 3. The Weak and Weak* Topologies
    3.1 Topological Vector Spaces
    3.1.1 Definition and Examples
    3.1.2 Convex Sets
    3.1.3 Elementary Properties of the Weak Topology
    3.1.4 Elementary Properties of the Weak* Topology
    3.2 The Banach–Alaoglu Theorem
    3.2.1 The Separable Case
    3.2.2 Invariant Measures
    3.2.3 The General Case
    3.3 The Banach–Dieudonn´e Theorem
    3.4 The Eberlein–Smulyan Theorem
    3.5 The Kre?in–Milman Theorem
    3.6 Ergodic Theory
    3.6.1 Ergodic Measures
    3.6.2 Space and Times Averages
    3.6.3 An Abstract Ergodic Theorem
    3.7 Problems

  • 4. Fredholm Theory
    4.1 The Dual Operator
    4.1.1 Definition and Examples
    4.1.2 Duality
    4.1.3 The Closed Image Theorem
    4.2 Compact Operators
    4.3 Fredholm Operators
    4.4 Composition and Stability
    4.5 Problems

  • 5. Spectral Theory
    5.1 Complex Banach Spaces
    5.1.1 Definition and Examples
    5.1.2 Integration
    5.1.3 Holomorphic Functions
    5.2 The Spectrum
    5.2.1 The Spectrum of a Bounded Linear Operator
    5.2.2 The Spectral Radius
    5.2.3 The Spectrum of a Compact Operator
    5.2.4 Holomorphic Functional Calculus
    5.3 Operators on Hilbert Spaces
    5.3.1 Complex Hilbert Spaces
    5.3.2 The Adjoint Operator
    5.3.3 The Spectrum of a Normal Operator
    5.3.4 The Spectrum of a Self-Adjoint Operator
    5.4 The Spectral Mapping Theorem.
    5.4.1 C* Algebras
    5.4.2 The Stone–Weierstraß Theorem
    5.4.3 Functional Calculus for Self-Adjoint Operators
    5.5 Spectral Representations
    5.5.1 The Gelfand Representation
    5.5.2 C* Algebras of Normal Operators
    5.5.3 Functional Calculus for Normal Operators
    5.6 Spectral Measures
    5.6.1 Projection Valued Measures
    5.6.2 Measurable Functional Calculus
    5.7 Cyclic Vectors
    5.8 Problems
  • 6. Unbounded Operators
    6.1 Unbounded Operators on Banach Spaces
    6.1.1 Definition and Examples
    6.1.2 The Spectrum of an Unbounded Operator
    6.1.3 Spectral Projections
    6.2 The Dual of an Unbounded Operator
    6.3 Unbounded Operators on Hilbert Spaces
    6.3.1 The Adjoint of an Unbounded Operator
    6.3.2 Unbounded Self-Adjoint Operators
    6.3.3 Unbounded Normal Operators
    6.4 Functional Calculus
    6.5 Spectral Measures
    6.6 Problems
  • 7. Semigroups of Operators
    7.1 Strongly Continuous Semigroups
    7.1.1 Definition and Examples
    7.1.2 Basic Properties
    7.1.3 The Infinitesimal Generato
    7.2 The Hille–Yosida–Phillips Theorem
    7.2.1 Well-Posed Cauchy Problems
    7.2.2 The Hille–Yosida–Phillips Theorem
    7.2.3 Contraction Semigroups
    7.3 Semigroups and Duality
    7.3.1 Banach Space Valued Measurable Functions
    7.3.2 The Banach Space Lp(I, X)
    7.3.3 The Dual Semigroup
    7.3.4 Semigroups on Hilbert Spaces
    7.4 Analytic Semigroups
    7.4.1 Properties of Analytic Semigroups
    7.4.2 Generators of Analytic Semigroups
    7.4.3 Examples of Analytic Semigroups
    7.5 Problems

  • A The Lemma of Zorn

  • B Tychonoff ’s Theorem

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