Now revised and updated, this brisk introduction to functional analysis is intended for advanced undergraduate students, typically final year, who have had some background in real analysis. The author's aim is not just to cover the standard material in a standard way, but to present results of application in contemporary mathematics and to show the relevance of functional analysis to other areas. Unusual topics covered include the geometry of finite-dimensional spaces, invariant subspaces, fixed-point theorems, and the Bishop-Phelps theorem. An outstanding feature is the large number of exercises, some straightforward, some challenging, none uninteresting.
Author(s): Béla Bollobás
Series: Cambridge Mathematical Textbooks
Edition: 2
Publisher: Cambridge University Press
Year: 1999
Language: English
Pages: 251
Cover ......Page 1
Title ......Page 2
Contents ......Page 6
Preface ......Page 8
1. Basic inequalities ......Page 11
2. Normed spaces and bounded linear operators ......Page 28
3. Linear functional and the Hahn-Banach theorem ......Page 55
4. Finite-dimensional normed spaces ......Page 70
5. The Baire category theorem and the closed-graph theorem ......Page 85
6. Continuous functions on compact spaces and the Stone-Weierstrass theorem ......Page 95
7. The contraction-mapping theorem ......Page 111
8. Weak topologies and duality ......Page 124
9. Euclidean spaces and Hilbert spaces ......Page 140
10. Orthonormal systems ......Page 151
11. Adjoint operators ......Page 165
12. The algebra of bounded linear operators ......Page 177
13. Compact operators on Banach spaces ......Page 196
14. Compact normal operators ......Page 208
15. Fixed-point theorems ......Page 223
16. Invariant subspaces ......Page 236
Index of notation ......Page 243
Index of terms ......Page 245
Back ......Page 251
