UID:
almafu_9960119319302883
Format:
1 online resource (239 pages) :
,
digital, PDF file(s).
ISBN:
1-107-71048-0
,
1-107-71303-X
,
1-139-17201-8
Series Statement:
Cambridge mathematical textbooks An introduction to Hilbert space
Content:
This textbook is an introduction to the theory of Hilbert space and its applications. The notion of Hilbert space is central in functional analysis and is used in numerous branches of pure and applied mathematics. Dr Young has stressed applications of the theory, particularly to the solution of partial differential equations in mathematical physics and to the approximation of functions in complex analysis. Some basic familiarity with real analysis, linear algebra and metric spaces is assumed, but otherwise the book is self-contained. It is based on courses given at the University of Glasgow and contains numerous examples and exercises (many with solutions). Thus it will make an excellent first course in Hilbert space theory at either undergraduate or graduate level and will also be of interest to electrical engineers and physicists, particularly those involved in control theory and filter design.
Note:
Title from publisher's bibliographic system (viewed on 05 Oct 2015).
,
Cover -- Half-title -- Title -- Copyright -- Contents -- Foreword -- Introduction -- 1 Inner product spaces -- 1.1 Inner product spaces as metric spaces -- 1.2 Problems -- 2 Normed spaces -- 2.1 Closed linear subspaces -- 2.2 Problems -- 3 Hilbert and Banach spaces -- 3.1 The space L[sup(2)](a, b) -- 3.2 The closest point property -- 3.3 Problems -- 4 Orthogonal expansions -- 4.1 Bessel's inequality -- 4.2 Pointwise and L[sup(2)] convergence -- 4.3 Complete orthonormal sequences -- 4.4 Orthogonal complements -- 4.5 Problems -- 5 Classical Fourier series -- 5.1 The Fejér kernel -- 5.2 Fejér's theorem -- 5.3 Parseval's formula -- 5.4 Weierstrass' approximation theorem -- 5.5 Problems -- 6 Dual spaces -- 6.1 The Riesz-Fréchet theorem -- 6.2 Problems -- 7 Linear operators -- 7.1 The Banach space L(E, F) -- 7.2 Inverses of operators -- 7.3 Adjoint operators -- 7.4 Hermitian operators -- 7.5 The spectrum -- 7.6 Infinite matrices -- 7.7 Problems -- 8 Compact operators -- 8.1 Hilbert-Schmidt operators -- 8.2 The spectral theorem for compact Hermitian operators -- 8.3 Problems -- 9 Sturm-Liouville systems -- 9.1 Small oscillations of a hanging chain -- 9.2 Eigenfunctions and eigenvalues -- 9.3 Orthogonality of eigenfunctions -- 9.4 Problems -- 10 Green's functions -- 10.1 Compactness of the inverse of a Sturm-Liouville operator -- 10.2 Problems -- 11 Eigenfunction expansions -- 11.1 Solution of the hanging chain problem -- 11.2 Problems -- 12 Positive operators and contractions -- 12.1 Operator matrices -- 12.2 Möbius transformations -- 12.3 Completing matrix contractions -- 12.4 Problems -- 13 Hardy spaces -- 13.1 Poisson's kernel -- 13.2 Fatou's theorem -- 13.3 Zero sets of H[sup(2)] functions -- 13.4 Multiplication operators and infinite Toeplitz and Hankel matrices -- 13.5 Problems -- 14 Interlude: complex analysis and operators in engineering.
,
15 Approximation by analytic functions -- 15.1 The Nehari problem -- 15.2 Hankel operators -- 15.3 Solution of Nehari's problem -- 15.4 Problems -- 16 Approximation by meromorphic functions -- 16.1 The singular values of an operator -- 16.2 Schmidt pairs and singular vectors -- 16.3 The Adamyan-Arov-Krein theorem -- 16.4 Problems -- Appendix: square roots of positive operators -- References -- Answers to selected problems -- Afterword -- Index of notation -- Subject index.
,
English
Additional Edition:
ISBN 0-521-33717-8
Additional Edition:
ISBN 0-521-33071-8
Language:
English
URL:
https://doi.org/10.1017/CBO9781139172011
