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\begin{document}
\begin{center}
Compact Sets, Operator norms, Compact operators
\end{center}

\noindent
In the following definitions and theorems, $H$ is a Hilbert space
though most definitions hold for general metric and/or normed
vector spaces.

\vspace{0.2in}
\noindent
{\bf Compact Sets}

Let $S \subset H$ be some set. Then
\vspace{0.2in}

\begin{tabular}{lll}
(C1) $S$ bounded & $\Leftrightarrow$ & 
$\exists M>0$ s.t. $\parallel x \parallel \le M , \forall x \in S$ \\
(C2) $S$ compact & $\Leftrightarrow$ &
Every sequence $\{ x_n \} \subset S$ contains \\
\ & \ & 
a convergent subsequence $\{x_{n_k}\}$ \\
 & \ & which converges to $x\in S$ \\
(C3) $S$ bounded & $\not{\Rightarrow}$ & $S$ compact \\
(C4) $S$ (sequentially) compact & $\Rightarrow$ & $S$ closed and bounded \\
(C5) $S\equiv \{x\in H : \parallel x \parallel \le 1\}$ compact
& $\Rightarrow$ & $dim(H) < \infty$
\end{tabular}



\vspace{0.2in}
\noindent
{\bf Definition: Bounded Operator} An operator $L:H \rightarrow H$
is bounded if there exists some $M>0$ such that
\eq
\parallel L x \parallel \le M \parallel x \parallel  \quad 
\forall x\in H
\endeq

\vspace{0.2in}
\noindent
{\bf Definition: Operator norm} For any bounded operator
$L:H\rightarrow H$ we define the operator norm as
\eq
\parallel L \parallel_{op}
\equiv \max_{\parallel x\parallel = 1} \parallel Lx \parallel
= \max_{x\ne 0}
\frac{\parallel Lx \parallel}{\parallel x \parallel}
\endeq



\vspace{0.2in}
\noindent
{\bf Definition: Compact  Operator} An operator $K:H\rightarrow H$
is compact if it transforms bounded sets into compact sets.

\vspace{0.3in}
\noindent
{\bf Compact and Bounded linear operators}

\vspace{0.3in}
\begin{tabular}{lll}
(CO1) $K$ compact & $\Rightarrow$ & $K$ bounded \\
(CO2) $K$ linear, $dim(R(K))<\infty$ & $\Rightarrow$ & $K$ compact \\
(CO3) $K$ bounded, $\{\phi_n\}_{n=1}^{\infty}$ orthonormal &
$\Rightarrow$ & $\lim_{N\rightarrow \infty} K\phi_n = 0$ \\
(CO4) $K_n$ compact, $\parallel K_n -K\parallel_{op} \rightarrow  0$
& $\Rightarrow$ & $K$ compact \\
(CO5) $K$ compact & $\Leftrightarrow$ &
$\{x_n\}\subset H$ bounded $\Rightarrow$ \\
\ & \ &  $\{Lx_n\}$ has a convergent subsequence \\
(CO6) $K_1,K_2$ compact & $\Rightarrow$ &
$K_1+K_2$ compact
\end{tabular}

\vspace{0.2in}
\noindent
Combining (CO4) and (CO6) we see the space of
compact operators is a closed linear space using
the operator norm.
\end{document}