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An Advanced Complex Analysis Problem Book: Topological Vector Spaces, Functional Analysis, and Hilbert Spaces of Analytic Functions

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This is an exercises book at the beginning graduate level, whose aim is to illustrate some of the connections between functional analysis and the theory of functions of one variable. A key role is played by the notions of positive definite kernel and of reproducing kernel Hilbert space. A number of facts from functional analysis and topological vector spaces are surveyed. Then, various Hilbert spaces of analytic functions are studied.

Author(s): Daniel Alpay
Publisher: Birkhäuser
Year: 2015

Language: English
Pages: 523
Tags: Functional Analysis; Operator Theory

Front Matter....Pages i-xviii
Front Matter....Pages 9-9
Algebraic Prerequisites....Pages 11-60
Analytic Functions....Pages 61-118
Front Matter....Pages 119-119
Topological Spaces....Pages 121-198
Normed Spaces....Pages 199-247
Locally Convex Topological Vector Spaces....Pages 249-283
Some Functional Analysis....Pages 285-327
Front Matter....Pages 329-329
Positive Definite Functions and Kernels, and Reproducing Kernel Hilbert Spaces....Pages 331-403
Hardy Spaces....Pages 405-441
de Branges–Rovnyak Spaces....Pages 443-459
Bergman Spaces....Pages 461-474
Fock Spaces....Pages 475-486
Back Matter....Pages 487-521