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\begin{document}

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\section*{operator topology}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

\hypertarget{functional_analysis}{}\paragraph*{{Functional analysis}}\label{functional_analysis}

[[!include functional analysis - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_norm_topology}{Relation to norm topology}\dotfill \pageref*{relation_to_norm_topology} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

An \textbf{operator topology} is an abbreviation of a \textbf{[[topology]] on a space of (continuous linear) [[linear operator|operator]]s} between [[topological vector space]]s over a fixed [[field]] $k$ of [[real number|reals]] or [[complex number|complexes]] (possibly also p-adics, skewfield of [[quaternions]] etc.). In other words the [[hom-sets]] in the category of topological vector spaces as objects and continuous linear operators as morphisms are equipped with an operator topology.

There are many widely used topologies, some with standard names. Let $L(V,W) = Hom_{TVS}(V,W)$ be the set of continuous linear operators.

\begin{itemize}%
\item \textbf{weak operator topology} on $L(V,W)$ is given by the basis of open neighborhoods of zero given by sets of the form $U(x,f) = \{A\in L(V,W) : |f(A(x)) \lt 1 \}$ where $x\in V$ and $f\in W^* = Hom_{TVS}(W,k)$. A sequence $(A_n)$ converges to $A$ in weak operator topology iff the sequence $(A_n(x))$ converges to $A(x)$ in the weak topology on $W$. We write $A_n\stackrel{w}\longrightarrow A$ or $w-lim A_n = A$.


\item \textbf{strong operator topology}: the basis of neighborhoods of zero is given by sets $N(x,U) = \{A\in L(V,W) \,|\, A v \in U\}$, where $v\in V$ and $U$ is a neighborhood of zero in $W$. For convergence of sequences, we write $A_n\stackrel{s}\longrightarrow A$ or $s-lim A_n= A$.


\item \textbf{uniform operator topology}: here we assume that $V,W$ are normed spaces with norms $p_V$, $p_W$. Then $L(V,W)$ has a uniform operator topology induced by the norm given by the formula



\end{itemize}
\begin{displaymath}
p(A) = sup_{v\neq 0} \frac{p_W(A v)}{p_V (v)}
\end{displaymath}
\begin{itemize}%
\item \textbf{ultraweak operator topology} \ldots{}

\end{itemize}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{relation_to_norm_topology}{}\subsubsection*{{Relation to norm topology}}\label{relation_to_norm_topology}

The reason that in the definition of a [[unitary representation]], the strong operator topology on $\mathcal{U}(\mathcal{H})$ is used and not the [[norm topology]], is that only few [[homomorphism]]s turn out to be [[continuous map|continuous]] in the norm topology.

Example: let $G$ be a [[compact topological space|compact]] [[Lie group]] and $L^2(G)$ be the [[Hilbert space]] of square integrable [[measurable function]]s with respect to its [[Haar measure]]. The right [[regular representation]] of $G$ on $L^2(G)$ is defined as

\begin{displaymath}
R: G \to \mathcal{U}(L^2(G))
\end{displaymath}
\begin{displaymath}
g \mapsto (R_g: f(x) \mapsto f(x g))
\end{displaymath}
and this will generally not be continuous in the norm topology, but is always continuous in the strong topology.

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item A. A. Kirillov, A. D. Gvi\v{s}iani,      (theorems and exercises in functional analysis), Moskva, Nauka 1979, 1988

\end{itemize}
[[!redirects topology on a space of operators]] [[!redirects strong operator topology]] [[!redirects weak operator topology]] [[!redirects uniform operator topology]]



\end{document}