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Hilbert Spaces - Sequel to Linear Algebra
Ryan Tully-Doyle
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Front Matter
Preface
1
Introduction
1.1
Motivation
1.2
Inner products
1.3
Basis and coordinates
1.4
Operators
2
Inner product spaces
2.1
Inner products and
\(\ell^2\)
2.2
Inner products, norms, and metric spaces
2.3
Examples of inner product spaces
2.4
Exercises
2.4
Exercises
3
Normed spaces
3.1
Norms
3.2
Closed linear subspaces
3.3
Exercises
3.3
Exercises
4
Hilbert and Banach spaces
4.1
Limits
4.2
\(L^2[a,b]\)
4.3
The closest point property
4.4
Exercises
4.4
Exercises
5
Orthogonal expansions
5.1
Geometry in Hilbert space
5.2
Bessel’s inequality
5.3
Modes of convergence
5.4
Complete orthonormal sequences
5.5
Orthogonal complements
5.6
Exercises
5.6
Exercises
6
Fourier Series
6.1
Convolution
6.2
The Fejér kernel
6.3
Convergence of classical Fourier series
6.4
Exercises
6.4
Exercises
7
Dual spaces
7.1
Linear functionals
7.2
The Riesz representation theorem
7.3
Exercises
8
Linear operators
8.1
Linear operators
8.2
Inverses
8.3
Adjoint operators
8.4
Hermitian operators
9
Appendix: Results from analysis
9.1
Results from real analysis
9.2
Results from complex analysis
Section
9.2
Results from complex analysis
Eventually necessary things will go here.
