\contentsline {part}{I\hspace {1em}Fundamentals}{1}{}%
\contentsline {chapter}{\numberline {1}Topological linear spaces}{3}{}%
\contentsline {section}{\numberline {1.1}Banach spaces}{4}{}%
\contentsline {section}{\numberline {1.2}Hilbert space}{4}{}%
\contentsline {chapter}{\numberline {2}Infinite dimensional linear spaces}{7}{}%
\contentsline {section}{\numberline {2.1}Infinite linear combinations}{7}{}%
\contentsline {section}{\numberline {2.2}Closed linear span}{7}{}%
\contentsline {section}{\numberline {2.3}Schauder basis (functional analysis)}{8}{}%
\contentsline {section}{\numberline {2.4}$\ell ^p$ spaces}{9}{}%
\contentsline {subsection}{\numberline {2.4.1}$p$-norm}{9}{}%
\contentsline {subsection}{\numberline {2.4.2}$\ell ^p$ space}{10}{}%
\contentsline {section}{\numberline {2.5}$L^p$ spaces}{10}{}%
\contentsline {subsection}{\numberline {2.5.1}Othogonal Schauder basis of an $L^p$ spaces}{12}{}%
\contentsline {subsection}{\numberline {2.5.2}Riesz-Fischer theorems}{13}{}%
\contentsline {section}{\numberline {2.6}$C^k$ spaces}{13}{}%
\contentsline {chapter}{\numberline {3}Operator theory}{15}{}%
\contentsline {section}{\numberline {3.1}Operator norms}{16}{}%
\contentsline {subsection}{\numberline {3.1.1}Matrix norms}{17}{}%
\contentsline {section}{\numberline {3.2}Uniform bounded principle}{18}{}%
\contentsline {section}{\numberline {3.3}Spectral theorem}{18}{}%
\contentsline {section}{\numberline {3.4}Hahn-Banach theorem}{18}{}%
\contentsline {section}{\numberline {3.5}Open mapping theorem}{18}{}%
\contentsline {section}{\numberline {3.6}Closed graph theorem}{18}{}%
\contentsline {section}{\numberline {3.7}Uniform bounded principle}{18}{}%
\contentsline {chapter}{\numberline {4}Banach algebrae}{19}{}%
\contentsline {section}{\numberline {4.1}Stone-Weierstrass theorem}{19}{}%
\contentsline {chapter}{\numberline {5}Banach and Hilbert spaces}{21}{}%
\contentsline {section}{\numberline {5.1}Hilbert spaces}{21}{}%
\contentsline {subsection}{\numberline {5.1.1}Projections on Hilbert spaces}{22}{}%
\contentsline {section}{\numberline {5.2}Banach spaces}{23}{}%
\contentsline {part}{II\hspace {1em}Advanced}{25}{}%
\contentsline {chapter}{\numberline {6}$W^*$-algebrae}{27}{}%
\contentsline {chapter}{\numberline {7}Spectral theory}{29}{}%
\contentsline {chapter}{\numberline {8}More function spaces}{31}{}%
\contentsline {section}{\numberline {8.1}$H^p$ space}{31}{}%
\contentsline {section}{\numberline {8.2}Fréchet space (functional analysis)}{31}{}%