An introduction to functional analysis / James C. Robinson

Table of Contents


PART I : Preliminaries

1. Vector Spaces and Bases
1.1 Definition of a vector space
1.2 Examples of vector spaces
1.3 Linear subspaces
1.4 Spanning sets, linear independence, and bases
1.5 Linear maps between vector spaces and their inverses
1.6 Existence of bases and zorn's Lemma

2. Metric Spaces
2.1 Metric spaces
2.2 Open and closed sets
2.3 Continuity and sequential continuity
2.4 Interior, closure, density, and separability
2.5 Compactness

PART II : Normed Linear Spaces

3. Norms and Normed Spaces
3.1 Norms
3.2 Examples of normed spaces
3.3 Convergence in normed spaces
3.4 Equivalent noems
3.5 Isomorphisms between normed spaces
3.6 Separability of normed spaces

4. Complete Normed Spaces
4.1 Banach spaces
4.2 Examples of banach spaces
4.3 Sequences in banach spaces
4.4 The contraction mapping theorem

5. Finite-Dimensional Normed Spaces
5.1 Equivalence of norms on finite-dimensional spaces
5.2 Compactness of the closed unit ball

6. Spaces of Continuous Functions
6.1 The Weierstrass Approximation Theorem
6.2 The stone- Weirstrass Theorem
6.3 The Arzela-Ascoli Theorem

7. Completions and the Lebesque Spaces
7.1 Non- Completeness of c ([0,1]) with the L Norm
7.2 The Completion of a normed spaces
7.3 Definitionof the L Spaces as Completions

PART III : Hilbert Spaces

8. Hilbert Spaces
8.1 Inner Products
8.2 The Cauchy- Schwarz inequality
8.3 Properties of the induced norms
8.4 Hilbert spaces

9. Orthonormal sets and orthonormal bases for hilbert spaces
9.1 Schauder bases in normed spaces
9.2 Orthonormal sets
9.3 Convergence of orthogonal series
9.4 Orthonormal bases for hilbert spaces
Separable Hilbert spaces

10. Closet Points and Approximation
10.1 Closest points in convex subsets of Hilbert spaces
10.2 Linear subspaces and Orthogonal complements
10.3 Best approximations

11. Linear Maps between Normed Spaces
11.1 Bounded linear maps
11.2 Some examples of bounded linear maps
11.3 Completeness of B (X,Y) when Y is complete
11.4 Kernel and range
11.5 Inverses and Invertibility

12. Dual Spaces and the Riesz Representation Theorem
12.1 The Dual Space
12.2 The Riesz Representation Theorem

13. The Hilbert Adjoint of a Linear Operator
14. The Spectrum of Bounded Linear Operator
15. Compact Linear Operator
16. The Hilbert-Schmidt Theorem
17. Application : Sturm-Liouville Problems

PART IV : Banach Spaces
18. Dual Spaces of Banach Spaces
19. The Hahn-Banach Theorem
20. Some Applications of the Hahn-Banach Theorem
21. Convex Subsets of Banach Spaces
22. The Principle of Uniform Boundedness
23. The Open Mapping, Inverse Mapping, and Closed Graph Theorems
24. Spectral Theory for Compact Operators
25. Unbounded Operators on Hilbert Spaces
26. Reflexive Spaces
27. Weak and Weak.* Convergence

Includes bibliographical references p. 394 and index p. 395-304