Functional analysis : spectral theory
著者
書誌事項
Functional analysis : spectral theory
（Birkhäuser advanced texts : Basler Lehrbücher / edited by Herbert Amann, Hanspeter Kraft）
Birkhäuser Verlag, c1997
- : us
- : sz
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注記
Includes bibliographical references (p. 237) and index
内容説明・目次
内容説明
In an elegant and concise fashion, this book presents the concepts of functional analysis required by students of mathematics and physics. It begins with the basics of normed linear spaces and quickly proceeds to concentrate on Hilbert spaces, specifically the spectral theorem for bounded as well as unbounded operators in separable Hilbert spaces. While the first two chapters are devoted to basic propositions concerning normed vector spaces and Hilbert spaces, the third chapter treats advanced topics which are perhaps not standard in a first course on functional analysis. It begins with the Gelfand theory of commutative Banach algebras, and proceeds to the Gelfand-Naimark theorem on commutative C*-algebras. A discussion of representations of C*-algebras follows, and the final section of this chapter is devoted to the Hahn-Hellinger classification of separable representations of commutative C*-algebras. After this detour into operator algebras, the fourth chapter reverts to more standard operator theory in Hilbert space, dwelling on topics such as the spectral theorem for normal operators, the polar decomposition theorem, and the Fredholm theory for compact operators.
A brief introduction to the theory of unbounded operators on Hilbert space is given in the fifth and final chapter. There is a voluminous appendix whose purpose is to fill in possible gaps in the reader's background in various areas such as linear algebra, topology, set theory and measure theory. The book is interspersed with many exercises, and hints are provided for the solutions to the more challenging of these.
目次
- Normed spaces - vector spaces, normed spaces, linear operators, the Hahn-Banach theorem, completeness, some topological considerations
- Hilbert spaces - inner product spaces, some preliminaries, orthonormal bases, the adjoint operator, strong and weak convergence
- C*-algebras - Banach algebras, Gelfand-Naimark theory, commutative C*-algebras, representations of C*-algebras, the Hahn-Hellinger theorem
- some operator theory - the spectral theorem, polar decomposition, compact operators,
- Fredholm operators and index
- unbounded operators - closed operators, symmetric and self-adjoint operators, spectral theorem and polar decomposition. Appendices: some linear algebra
- transfinite considerations
- topological spaces
- compactness
- measure and integration
- the Stone-Weierstrass theorem
- the Riesz representation theorem.
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