泛函分析-(影印版)

.4.1 extension of positive linear functionals, 29
4.2 banach limits. 31
4.3 finitely additive invariant set functions, 33
historical note, 34

5. normed linear spaces
5.1 norms, 36
norms for quotient spaces-complete normed linear spaces-the spaces c, b-lp spaces and h61der's inequality-sobolev spaces, embedding theorems-separable spaces
5.2 noncompactness of the unit bail, 43
uniform convexity-the mazur-ulam theorem on isometrics
5.3 isometrics, 47

6. hilbert space
6.1 scalar product, 52
schwarz inequality parallelogram identity——completeness,closure-e2, l2
6.2 closest point in a closed convex subset, 54orthogonal complement of a subspace-orthogonal decomposition
6.3 linear functionals, 56
the riesz-frechet representation theorem-lax-milgram lemma
6.4 linear span, 58
orthogonal projection-orthonormal bases, gram-schmidt process-isometries of a hilbert space

7. applications of hilbert space results
7.1 radon-nikodym theorem, 63
7.2 dirichlet's problem, 65
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, 72
dual space
8.2 extension of bounded linear functionals, 74
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, 78
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, 83
dual characterization of convex hull-dual characterization of distance from a closed, convex set

9. applications of duality
9.1 completeness of weighted powers, 87
9.2 the muntz approximation theorem, 88
9.3 runge'stheorem, 91
9.4 dual variational problems in function theory, 91
9.5 existence of green's function, 94

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 helly's theorem

11. applications of weak convergence
11.1 approximation of the function by continuous functions, 108 toeplitz's theorem on summability
11.2 divergence of fourier series, 109
11.3 approximate quadrature, 110
11.4 weak and strong analyticity of vector-valued functions, 111
11.5 existence of solutions of partial differential equations, 112 galerkin's 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-alaoglu's theorem

13. locally convex topologies and the krein-milman theorem
13.1 separation of points by linear functionals, 123
13.2 the krein-milman theorem, 124
13.3 the stone-weierstrass theorem, 126
13.4 choquet's theorem, 128

14. examples of convex sets and their extreme points
14.1 positivefunctionals, 133
14.2 convex functions, 135
14.3 completely monotone functions, 137
14.4 theorems of caratheodory and bochner, 141
14.5 a theorem of krein, 147
14.6 positive harmonic functions, 148
14.7 the hamburger moment problem, 150
14.8 g. birkhoff's conjecture, 151
14.9 de finetti's theorem, 156
14.10 measure-preserving mappings, 157
historical note, 159

15. bounded linear maps
15.1 boundedness and continuity, 160
norm of a bounded linear map-transpose
15.2 strong and weak topologies, 165
strong and weak sequential convergence
15.3 principle of uniform boundedness, 166
15.4 composition of bounded maps, 167
15.5 the open mapping principle, 168
closed graph theorem historical note, 172

16. examples of bounded linear maps
16.1 boundedness of integral operators, 173
integral operators of hilbert-schmidt type-integral operators of holmgren type
16.2 the convexity theorem of marcel riesz, 177
16.3 examples of bounded integral operators, 180
the fourier transform, parseval's 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. zorn's lemma
author index
subject index