Foreword
1. Linear Spaces
Axioms for linear spaces-Infinite-dimensional examples-Subspace, linear span-Quotient space-Isomorphism-Convex sets-Extreme subsets

2. Linear Maps
2.1 Algebra of linear maps,
Axioms for linear maps-Sums and composites-Invertible linear maps-Nullspace and range-Invariant subspaces
2.2. Index of a linear map,
Degenerate maps-Pseudoinverse-IndexmProduct formula for the index-Stability of the index

3. The Hahn,Banach Theorem
3.1 The extension theorem,
Positive homogeneous, subadditive functionals-Extension of linear functionals-Gauge functions of convex sets
3.2 Geometric Hahn-Banach theorem,
The hyperplane separation theorem
3.3 Extensions of the Hahn-Banach theorem,
The Agnew-Morse theorem-The
Bohnenblust-Sobczyk-Soukhomlinov theorem

4. Applications of the Hahn-Banach theorem
4.1 Extension of positive linear functionals,
4.2 Banach limits.
4.3 Finitely additive invariant set functions,
Historical note,

5. Normed Linear Spaces
5.1 Norms,
Norms for quotient spaces-Complete normed linear spaces-The spaces C, B-Lp spaces and H61ders inequality-Sobolev spaces, embedding theorems-Separable spaces
5.2 Noncompactness of the unit bail,
Uniform convexity-The Mazur-Ulam theorem on isometrics
5.3 Isometrics,

6. Hilbert Space
6.1 Scalar product,
Schwarz inequality Parallelogram identity——Completeness,closure-e2, L
6.2 Closest point in a closed convex subset, 54Orthogonal complement of a subspace-Orthogonal decomposition
6.3 Linear functionals,
The Riesz-Frechet representation theorem-Lax-Milgram lemma
6.4 Linear span,
Orthogonal projection-Orthonormal bases, Gram-Schmidt process-Isometries of a Hilbert space

7. Applications of Hilbert Space Results
7.1 Radon-Nikodym theorem,
7.2 Dirichlets problem,
Use of the Riesz-Frechet theorem-Use of the Lax-Milgram theorem Use of orthogonal decomposition

8. Duals of Normed Linear Spaces
8.1 Bounded linear functionals,
Dual space
8.2 Extension of bounded linear functionals,
Dual characterization of norm-Dual characterization of distance from a subspace-Dual characterization of the closed linear span of a set
8.3 Reflexive spaces,
Reflexivity of Lp, 1 < p < -Separable spaces-Separability of the dual-Dual of C(Q), Q compact-Reflexivity of subspaces
8.4 Support function of a set,
Dual characterization of convex hull-Dual characterization of distance from a closed, convex set

9. Applications of Duality
9.1 Completeness of weighted powers,
9.2 The Muntz approximation theorem,
9.3 Rungestheorem,
9.4 Dual variational problems in function theory,
9.5 Existence of Greens function,

10. Weak Convergence
10.1 Uniform boundedness of weakly convergent sequences, 101 Principle of uniform boundedness-Weakly sequentially closed convex sets
10.2 Weak sequential compactness, 104 Compactness of unit ball in reflexive space
10.3 Weak* convergence, 105 Hellys theorem

11. Applications of Weak Convergence
11.1 Approximation of the function by continuous functions, 108 Toeplitzs theorem on summability
11.2 Divergence of Fourier series,
11.3 Approximate quadrature,
11.4 Weak and strong analyticity of vector-valued functions,
11.5 Existence of solutions of partial differential equations, 112 Galerkins method
11.6 The representation of analytic functions with positive real part, 115 Hergiotz-Riesz theorem

12. The Weak and Weak* Topologies
Comparison with weak sequential topology-Closed convex sets in the weak topology——Weak compactness-Alaoglus theorem

13. Locally Convex Topologies and the Krein-Milman Theorem
13.1 Separation of points by linear functionals,
13.2 The Krein-Milman theorem,
13.3 The Stone-Weierstrass theorem,
13.4 Choquets theorem,

14. Examples of Convex Sets and Their Extreme Points
14.1 Positivefunctionals,
14.2 Convex functions,
14.3 Completely monotone functions,
14.4 Theorems of Caratheodory and Bochner,
14.5 A theorem of Krein,
14.6 Positive harmonic functions,
14.7 The Hamburger moment problem,
14.8 G. Birkhoffs conjecture,
14.9 De Finettis theorem,
14.10 Measure-preserving mappings,
Historical note,

15. Bounded Linear Maps
15.1 Boundedness and continuity,
Norm of a bounded linear map-Transpose
15.2 Strong and weak topologies,
Strong and weak sequential convergence
15.3 Principle of uniform boundedness,
15.4 Composition of bounded maps,
15.5 The open mapping principle,
Closed graph theorem Historical note,

16. Examples of Bounded Linear Maps
16.1 Boundedness of integral operators,
Integral operators of Hilbert-Schmidt type-Integral operators of Holmgren type
16.2 The convexity theorem of Marcel Riesz,
16.3 Examples of bounded integral operators,
The Fourier transform, Parsevals theorem and Hausdorff-Young inequality-The Hilbert transform The Laplace transform-The Hilbert-Hankel transform
……
A. Riesz-Kakutani representation theorem
B. Theory of distributions
C. Zorns Lemma
Author Index
Subject Index