\contentsline {section}{\numberline {1}Metric Spaces}{3}
\contentsline {subsection}{\numberline {1.1}Open Sets, Closed Sets}{5}
\contentsline {subsection}{\numberline {1.2}Convergence, Cauchy Sequence, Completeness}{7}
\contentsline {subsection}{\numberline {1.3}Completeness Proofs}{9}
\contentsline {subsection}{\numberline {1.4}Completion of Metric Spaces}{10}
\contentsline {section}{\numberline {2}Normed Spaces}{13}
\contentsline {subsection}{\numberline {2.1}Finite dimensional Normed Spaces or Subspaces}{15}
\contentsline {subsection}{\numberline {2.2}Compactness and Finite Dimension}{18}
\contentsline {subsection}{\numberline {2.3}Linear Operators}{22}
\contentsline {subsection}{\numberline {2.4}Bounded and Continuous Linear Operators}{24}
\contentsline {subsection}{\numberline {2.5}Linear Functionals}{29}
\contentsline {subsection}{\numberline {2.6}Finite Dimensional Case}{32}
\contentsline {subsubsection}{\numberline {2.6.1}Linear Functionals}{33}
\contentsline {subsection}{\numberline {2.7}Dual Space}{35}
\contentsline {section}{\numberline {3}Hilbert Spaces}{39}
\contentsline {subsection}{\numberline {3.1}Representation of Functionals on Hilbert Spaces}{49}
\contentsline {subsection}{\numberline {3.2}Hilbert Adjoint}{53}
\contentsline {section}{\numberline {4}Fundamental Theorems}{59}
\contentsline {subsection}{\numberline {4.1}Bounded, Linear Functionals on $C[a,b]$}{68}
\contentsline {subsection}{\numberline {4.2}Adjoint Operator}{71}
\contentsline {subsection}{\numberline {4.3}Reflexive Spaces}{75}
\contentsline {subsection}{\numberline {4.4}Baire Category Theorem}{78}
\contentsline {subsection}{\numberline {4.5}Strong and Weak Convergence}{81}
\contentsline {subsection}{\numberline {4.6}The Open Mapping Theorem}{88}
\contentsline {subsection}{\numberline {4.7}Closed Graph Theorem}{89}
\contentsline {section}{\numberline {5}Exam 1}{93}
\contentsline {section}{\numberline {6}Exam 2}{95}
\contentsline {section}{\numberline {7}Final Exam}{96}