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
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