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Let $X$ be a normed vector space and $B(X)$ the space of bounded operators in $X$ . There are several interesting topologies that can be given to $B(X)$ . In what follows, $T_{\alpha}$ denotes a net in $B(X)$ and $T$ denotes an element of $B(X)$ .
Note: On 4, 5, 6 and 7, $X$ must be a Hilbert space.
This is the topology induced by the usual operator norm.
 in the norm topology  in the strong operator topology  in the weak operator topology $\,$
In case $X$ is an Hilbert space with inner product $\langle \cdot, \cdot \rangle$ , we have that
 in the weak operator topology The $\sigma$ -strong operator topology is the topology generated by the family of semi-norms $\|\cdot\|_S\;, S \in K(X)$ , defined by $\|T\|_S := \|TS\|$ . That means
 in the $\sigma$ -strong operator topology $\,$
Equivalently, $T_{\alpha} \longrightarrow T\;\; \Longleftrightarrow \; T_{\alpha}S \longrightarrow TS$ in norm for every $S \in K(X)$ .
This topology is also called the ultra-strong operator topology.
In this topology $X$ must be a Hilbert space. Let $B(X)_*$ denote the space of trace-class operators on $X$ and $Tr(S)$ the trace of an operator $S \in B(X)_*$ .
The $\sigma$ -weak operator topology is the topology generated by the family of semi-norms $\{\omega_{S} : S \in B(X)_*\}$ defined by $\omega_{S}(T) := |Tr(TS)|$ . That means
 in the $\sigma$ -weak operator topology![$\displaystyle \;\; \Longleftrightarrow\; \vert Tr[(T_{\alpha}-T)S]\vert \longrightarrow 0 \quad, \forall S \in B(X)_* $](http://images.planetmath.org:8080/cache/objects/9729/js/img12.png "$\displaystyle \;\; \Longleftrightarrow\; \vert Tr[(T_{\alpha}-T)S]\vert \longrightarrow 0 \quad, \forall S \in B(X)_* $") This topology is also called the ultra-weak operator topology.
In this topology $X$ must be a Hilbert space. In the following $T^*$ denotes the adjoint operator of $T$ .
The strong-* operator topology is the topology generated by the family of semi-norms $\| \cdot \|_{x}\;, x \in X$ defined by $\|T \|_{x} := \|Tx \|+\|T^*x\|$ . That means
 in the strong-* operator topology Equivalently, $T_{\alpha} \longrightarrow T$ if and only if $T_{\alpha}x \longrightarrow Tx$ and $T_{\alpha}^*x \longrightarrow T^*x$ , for every $x \in X$ .
In this topology $X$ must be a Hilbert space. Let $K(X)$ denote the space of compact operators on $X$ . In the following $T^*$ denotes the adjoint operator of $T$ .
The $\sigma$ -strong-* operator topology is the topology generated by the family of semi-norms $\| \cdot \|_S\;, S \in K(X)$ defined by $\|T \|_S := \|TS \|+\|T^*S\|$ . That means
 in the $\sigma$ -strong-* operator topology $\,$
Equivalently, $T_{\alpha} \longrightarrow T$ if and only if $T_{\alpha}S \longrightarrow TS$ and $T_{\alpha}^*S \longrightarrow T^*S$ in norm, for every $S \in K(X)$ .
This topology is also called ultra-strong-* operator topology.
- The norm topology is the strongest of the topologies defined above.
- The weak operator topology is weaker than the strong operator topology, which is weaker than the norm topology.
- In Hilbert spaces we can summarize the relations of the above topologies in the following diagram. Given two topologies $\mathcal{U},\mathcal{V}$ the notation $\mathcal{U} \rightarrow \mathcal{V}$ means $\mathcal{U}$ is weaker than $\mathcal{V}$ :
![$\displaystyle \xymatrix{ weak \ar[r] \ar[d] & strong \ar[r] \ar[d] & \emph{stro... ...r] & \emph{<SPAN class=](http://images.planetmath.org:8080/cache/objects/9729/js/img17.png "$\displaystyle \xymatrix{ weak \ar[r] \ar[d] & strong \ar[r] \ar[d] & \emph{stro... ...r] & \emph{<SPAN class=") $\sigma$ -strong} \ar[r] & \emph{ $\sigma$ -strong-*} \ar[r] & Norm} $">
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