Basic Lecture Notes

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\title{Lecture Notes: Functional Analysis}
\author{Compiled by First Last}
\date{\today}

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\section*{Preface}
These notes were compiled during the Fall semester and are continually revised.
Errors are my own; corrections welcome.

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\section{Lecture 1: Normed Spaces}
\begin{definition}
A \emph{norm} on a vector space $X$ over $\mathbb{R}$ or $\mathbb{C}$ is a map
$\|\cdot\|: X \to [0,\infty)$ satisfying positive-definiteness, homogeneity,
and the triangle inequality.
\end{definition}

\begin{example}
$\ell^p$ spaces are Banach spaces for $1 \le p \le \infty$.
\end{example}

\begin{theorem}[Equivalence of norms]
Any two norms on a finite-dimensional space induce the same topology.
\end{theorem}
\begin{proof}
By compactness of the unit sphere. Omitted here; see Rudin.
\end{proof}

\section{Lecture 2: Banach Spaces}
\begin{definition}
A normed space is a \emph{Banach space} if it is complete.
\end{definition}

\begin{theorem}[Banach--Steinhaus / Uniform Boundedness]
Let $X$ be a Banach space and $\{T_\alpha\}$ a family of bounded linear operators
on $X$. If $\sup_\alpha \|T_\alpha x\| < \infty$ for each $x$, then $\sup_\alpha \|T_\alpha\| < \infty$.
\end{theorem}

\begin{remark}
A major application: weak convergence implies boundedness.
\end{remark}

\section{Lecture 3: Hilbert Spaces}
\begin{definition}
A Hilbert space is a complete inner-product space.
\end{definition}

\begin{theorem}[Riesz Representation]
Every bounded linear functional on a Hilbert space $H$ is of the form
$\phi(x) = \langle x, y\rangle$ for a unique $y \in H$.
\end{theorem}

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