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Functional Analysis Topics


1. Classical Linear Spaces


2. Completeness, Linear Functionals, Linear Operators, and Compactness of the Closed Unit Ball

2.1. Completeness
2.2. Linear Functionals, Bounded Linear Functionals, and Dual Spaces
2.3. Linear Operators, Bounded Linear Operators, and the Space of Bounded Linear Operators
2.4. Isomorphisms and Isometries
2.5. Compactness of the Closed Unit Ball
2.6. Algebraic and Topological Complements of Linear Subspaces

3. The Open Mapping Theorem, Closed Graph Theorem, Uniform Boundedness Principle, and Hahn-Banach Theorem


4. The Weak Topology and Weak* Topology

4.1. Topological Spaces
4.2. The Weak and Weak-* Topologies
4.3. Reflexive Spaces
4.4. Helley's Theorem
4.5. Topological Vector Spaces
4.6. Separation
4.7. The Krein-Milman Theorem
4.8. Alaoglu's Theorem, Kakutani's Theorem, Goldstine's Theorem, and the Eberlein-Smulian Theorem

5. Inner Product Spaces

5.1. Inner Products and Inner Product Spaces
5.2. Hilbert Spaces
5.3. Orthogonal and Orthonormal Sets
5.4. Orthonormal Bases

6. Normed Algebras

6.1. Algebras, Normed Algebras, and Banach Algebras
6.2. Invertible and Singular Elements in an Algebra
6.3. Quasi-Invertible and Quasi-Singular Elements in an Algebra
6.4. The Spectrum of an Element in an Algebra
6.5. Ideals
6.6. Left X-Modules, Right X-Modules, and X-Bimodules
6.7. Approximate Identities
6.8. Commutative Subsets of an Algebra
6.9. Multiplicative Linear Functionals on a Banach Algebra
6.10 The Gelfand Representation of a Commutative Banach Algebra
6.11. Tensor Products
6.12. Amenability of Banach Algebras over C with Unit

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References
  • 1. Real Analysis (3rd Edition) by Halsey Royden.
  • 2. Real Analysis (4th Edition) by Halsey Royden and Patrick Fitzpatrick.
  • 3. Complete Normed Algebras by Frank F. Bonsall and John Duncan.
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