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Hilbert Spaces - Sequel to Linear Algebra
Ryan Tully-Doyle
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\(\DeclareMathOperator{\RE}{Re} \DeclareMathOperator{\IM}{Im} \DeclareMathOperator{\ess}{ess} \DeclareMathOperator{\intr}{int} \DeclareMathOperator{\dist}{dist} \DeclareMathOperator{\dom}{dom} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator{\cl}{cl} \DeclareMathOperator{\spn}{span} \DeclareMathOperator{\proj}{proj} \DeclareMathOperator\trace{trace} \DeclareMathOperator\re{\mathrm {Re~}} \DeclareMathOperator\im{\mathrm {Im~}} \newcommand\dd{\mathrm d} \newcommand{\eps}{\varepsilon} \newcommand{\To}{\longrightarrow} \newcommand{\hilbert}{\mathcal{H}} \newcommand{\HH}{\mathcal{H}} \newcommand{\s}{\mathcal{S}_2} \newcommand{\A}{\mathcal{A}} \newcommand\h{\mathcal{H}} \newcommand{\J}{\mathcal{J}} \newcommand{\K}{\mathcal{K}} \newcommand{\M}{\mathcal{M}} \newcommand{\F}{\mathbb{F}} \newcommand{\N}{\mathcal{N}} \renewcommand{\L}{\mathcal{L}} \newcommand{\T}{\mathbb{T}} \newcommand{\W}{\mathcal{W}} \newcommand{\X}{\mathcal{X}} \newcommand{\D}{\mathbb{D}} \newcommand{\C}{\mathbb{C}} \newcommand{\BOP}{\mathbf{B}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\BH}{\mathbf{B}(\mathcal{H})} \newcommand{\KH}{\mathcal{K}(\mathcal{H})} \newcommand{\pick}{\mathcal{P}_2} \newcommand{\schur}{\mathcal{S}_2} \newcommand{\R}{\mathbb{R}} \newcommand{\Complex}{\mathbb{C}} \newcommand{\Field}{\mathbb{F}} \newcommand{\RPlus}{\Real^{+}} \newcommand{\Polar}{\mathcal{P}_{\s}} \newcommand{\Poly}{\mathcal{P}(E)} \newcommand{\EssD}{\mathcal{D}} \newcommand{\Lop}{\mathcal{L}(\mathcal{H})} \newcommand{\cc}[1]{\overline{#1}} \newcommand{\abs}[1]{\left\vert#1\right\vert} \newcommand{\set}[1]{\left\{#1\right\}} \newcommand{\seq}[1]{\left\lt#1\right>} \newcommand{\norm}[1]{\lVert#1\rVert} \newcommand{\essnorm}[1]{\norm{#1}_{\ess}} \newcommand{\tr}{\operatorname{tr}} \newcommand{\ran}[1]{\operatorname{ran}#1} \newcommand{\nt}{\stackrel{\mathrm {nt}}{\to}} \newcommand{\pnt}{\xrightarrow{pnt}} \newcommand{\ip}[2]{\left\langle #1, #2 \right\rangle} \newcommand{\ad}{^\ast} \newcommand{\inv}{^{-1}} \newcommand{\adinv}{^{\ast -1}} \newcommand{\invad}{^{-1 \ast}} \newcommand\Pick{\mathcal P} \newcommand\Ha{\mathbb{H}} \newcommand\Htau{\mathbb{H}(\tau)} \newcommand{\vp}{\varphi} \newcommand{\ph}{\varphi} \newcommand\al{\alpha} \newcommand\ga{\gamma} \newcommand\de{\delta} \newcommand\ep{\varepsilon} \newcommand\la{\lambda} \newcommand\up{\upsilon} \newcommand\si{\sigma} \newcommand\beq{\begin{equation}} \newcommand\ds{\displaystyle} \newcommand\eeq{\end{equation}} \newcommand\df{\stackrel{\rm def}{=}} \newcommand\ii{\mathrm i} \newcommand{\vectwo}[2] { \begin{pmatrix} #1 \\ #2 \end{pmatrix} } \newcommand{\vecthree}[3] { \begin{pmatrix} #1 \\ #2 \\ #3 \end{pmatrix} } \newcommand\blue{\color{blue}} \newcommand\black{\color{black}} \newcommand\red{\color{red}} \newcommand\nn{\nonumber} \newcommand\bbm{\begin{bmatrix}} \newcommand\ebm{\end{bmatrix}} \newcommand\bpm{\begin{pmatrix}} \newcommand\epm{\end{pmatrix}} \numberwithin{equation}{section} \newcommand\nin{\noindent} \newcommand{\nCr}[2]{\,_{#1}C_{#2}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} \)
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
8.5
The spectrum
9
Compact operators
9.1
The spectral theorem for compact operators
10
Positive operators
10.1
Positive matrices
10.2
Positive operators in
\(\Lop\)
10.3
\(H^2\)
and
\(H^\infty(\D)\)
11
Appendix: Results from analysis
11.1
Results from real analysis
11.2
Results from complex analysis
12
Spectral theory
12.1
Banach algebra stuff
Section
11.2
Results from complex analysis
Eventually necessary things will go here.
