Introduction to the Analysis of Normed Linear Spaces

This is a basic course in functional analysis for senior undergraduate and beginning postgraduate students.

J. R. Giles (Author)

9780521653756, Cambridge University Press

Paperback, published 13 March 2000

296 pages, 19 b/w illus. 203 exercises
22.9 x 15.2 x 1.7 cm, 0.44 kg

'… the text is up-to-date and detailed in exposition, and is large enough in material covered for different courses to be constructed from it …'. Australian Mathematical Society Gazette

This text is a basic course in functional analysis for senior undergraduate and beginning postgraduate students. It aims at providing some insight into basic abstract analysis which is now the contextual language of much modern mathematics. Although it is assumed that the student will have familiarity with elementary real and complex analysis and linear algebra and have studied a course in the analysis of metric spaces, a knowledge of integration theory or general topology is not required. The theme of this text concerns structural properties of normed linear spaces in general, especially associated with dual spaces and continuous linear operators on normed linear spaces. But the implications of the general theory are illustrated with a great variety of example spaces.

1. Basic properties of normed linear spaces
2. Classes of example spaces
3. Orthonormal sets in inner product spaces
4. Norming mappings and forming duals and operator algebras
5. The shape of the dual
6. The Hahn–Banach theorem
7. The natural embedding and reflexivity
8. Subreflexivity
9. Baire category theory for metric spaces
10. The open mapping and closed graph theorems
11. The uniform boundedness theorem
12. Conjugate mappings
13. Adjoint operators on Hilbert space
14. Projection operators
15. Compact operators
16. The spectrum
17. The spectrum of a continuous linear operator
18. The spectrum of a compact operator
19. The spectral theorem for compact normal operators on Hilbert space
20. The spectral theorem for compact operators on Hilbert space
Appendices. A1. Zorn's lemma
A2. Numerical equivalence
A3. Hamel basis.

Subject Areas: Calculus & mathematical analysis [PBK]