Essential Results of Functional Analysis
168 pages

© 1990
Contents
Preface
0. Background
0.A. Review of basic functional analysis
0.B. Some special properties of integration in Rn
1. Topological vector spaces and operators
1.1. Examples of spaces
1.2. Examples of operators
1.3. Operator topologies and groups of operators
2. Convexity and fixed point theorems
2.1. KakutaniMarkov fixed point theorem
2.2. Haar measure for compact groups
2.3. KreinMillman theorem
3. Compact operators
3.1. Compact operators and HilbertSchmidt operators
3.2. Spectral theorem for compact normal operators
3.3. PeterWeyl theorem for compact groups
4. General spectral theory
4.1. Spectrum of an operator
4.2. Spectral theorem for selfadjoint operators
4.3. Gelfand's theory of commutative C*algebras
4.4. Mean ergodic theorem
5. Fourier transforms and Sobolev embedding theorems
5.1. Basic properties of the Fourier transform and the Plancherel theorem
5.2. Sobolev and Rellich embedding theorems
6. Distributions and elliptic operators
6.1. Basic properties of distributions
6.2. Distributions and Sobolev spaces
6.3. Regularity for elliptic operators
6.4. Appendix to 6.3: proof of Garding's inequality
6.5. A spectral theorem for elliptic operators
Index
0. Background
0.A. Review of basic functional analysis
0.B. Some special properties of integration in Rn
1. Topological vector spaces and operators
1.1. Examples of spaces
1.2. Examples of operators
1.3. Operator topologies and groups of operators
2. Convexity and fixed point theorems
2.1. KakutaniMarkov fixed point theorem
2.2. Haar measure for compact groups
2.3. KreinMillman theorem
3. Compact operators
3.1. Compact operators and HilbertSchmidt operators
3.2. Spectral theorem for compact normal operators
3.3. PeterWeyl theorem for compact groups
4. General spectral theory
4.1. Spectrum of an operator
4.2. Spectral theorem for selfadjoint operators
4.3. Gelfand's theory of commutative C*algebras
4.4. Mean ergodic theorem
5. Fourier transforms and Sobolev embedding theorems
5.1. Basic properties of the Fourier transform and the Plancherel theorem
5.2. Sobolev and Rellich embedding theorems
6. Distributions and elliptic operators
6.1. Basic properties of distributions
6.2. Distributions and Sobolev spaces
6.3. Regularity for elliptic operators
6.4. Appendix to 6.3: proof of Garding's inequality
6.5. A spectral theorem for elliptic operators
Index
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