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Full Description
The basic topics of the subject that every analyst should know are discussed in the first volume. This second part 'Select Topics' provides glimpses of some beautiful vistas of the functional analytic terrain that a reader may like to explore and enjoy. Introductions to these topics, including those of current interest, with pointers to further edification are provided. An author, with various constraints, necessarily faces a Hamletian dilemma in choosing the topics for such a work. The choices made here are: a little Fourier analysis (Lp, Schwartz and Sobolev spaces, Fourier transform and tempered distributions, interpolation, spectral synthesis), some operator theory (compact, Hilbert-Schmidt, trace class, Fredholm and Toeplitz operators, spectral theorem, invariant subpaces), operator algebras (commutative Banach algebras, C∗algebras, von Neumann algebras and operator spaces), Gabor analysis (Hilbert frames, Gabor frames, Riesz bases, wavelets), Banach space geometry, Schauder bases and functional analytic methods in PDE (scattered throughout the book). It also gives some unusual applications. Several mini courses and projects are possible based on the book, besides an advanced course in Functional Analysis. A historical perspective blending harmoniously with the flow, a clear succinct style that is not terse nor verbose and biographic thumbnails continue. The bibliography is extensive and includes all original sources.
Contents
Chapter 1 Function spaces 2: Lp-spaces.- Chapter 2 Function spaces 3: Schwartz and Sobolev spaces.- Chapter 3 Fourier transform and tempered distributions.- Chapter 4 Interpolation of Operators.- Chapter 5 Compact operators 3 - Riesz-Schauder theory.- Chapter 6 Hilbert-Schmidt Operators.- Chapter 7 Trace class and nuclear operators.- Chapter 8 Spectral theorem - a revisit.- Chapter 9 Commutative Banach algebras.- Chapter 10 C∗-algebras.- Chapter 11 von Neumann algebras.- Chapter 12 Operator spaces.- Chapter 13 Weak compactness.- Chapter 14 A taste of Banach space geometry.- Chapter 15 The Laplace operator.- Chapter 16 Hilbert frames.- Chapter 17 Weyl-Heisenberg frames.- Chapter 18 Invariant subspaces.- Chapter 19 Fredholm and Toeplitz operators.- Chapter 20 Ideals in L1(Rn) and Spectral synthesis.
