Editing Weak operator topology (section)

==SOT and WOT on ''B(X,Y)'' when ''X'' and ''Y'' are normed spaces==

We can extend the definitions of SOT and WOT to the more general setting where ''X'' and ''Y'' are [[normed vector space|normed spaces]] and <math>B(X,Y)</math> is the space of bounded linear operators of the form <math>T:X\to Y</math>.  In this case, each pair <math>x\in X</math> and <math>y^*\in Y^*</math> defines a [[norm (mathematics)|seminorm]] <math>\|\cdot\|_{x,y^*}</math> on <math>B(X,Y)</math> via the rule <math>\|T\|_{x,y^*}=|y^*(Tx)|</math>.  The resulting family of seminorms generates the '''weak operator topology''' on <math>B(X,Y)</math>.  Equivalently, the WOT on <math>B(X,Y)</math> is formed by taking for [[base (topology)|basic open neighborhoods]] those sets of the form

:<math>N(T,F,\Lambda,\epsilon):= \left \{S\in B(X,Y): \left |y^*((S-T)x) \right |<\epsilon,x\in F,y^*\in\Lambda \right \},</math>

where <math>T\in B(X,Y), F\subseteq X</math> is a finite set, <math>\Lambda\subseteq Y^*</math> is also a finite set, and <math>\epsilon>0</math>.  The space <math>B(X,Y)</math> is a locally convex topological vector space when endowed with the WOT.

The '''strong operator topology''' on <math>B(X,Y)</math> is generated by the family of seminorms <math>\|\cdot\|_x, x\in X,</math> via the rules <math>\|T\|_x=\|Tx\|</math>.  Thus, a topological base for the SOT is given by open neighborhoods of the form

:<math>N(T,F,\epsilon):=\{S\in B(X,Y):\|(S-T)x\|<\epsilon,x\in F\},</math>

where as before <math>T\in B(X,Y), F\subseteq X</math> is a finite set, and <math>\epsilon>0.</math>

===Relationships between different topologies on ''B(X,Y)''===

The different terminology for the various topologies on <math>B(X,Y)</math> can sometimes be confusing.  For instance, "strong convergence" for vectors in a normed space sometimes refers to norm-convergence, which is very often distinct from (and stronger than) than SOT-convergence when the normed space in question is <math>B(X,Y)</math>.  The [[weak topology]] on a normed space <math>X</math> is the coarsest topology that makes the linear functionals in <math>X^*</math> continuous; when we take <math>B(X,Y)</math> in place of <math>X</math>, the weak topology can be very different than the weak operator topology.  And while the WOT is formally weaker than the SOT, the SOT is weaker than the operator norm topology.

In general, the following inclusions hold:

:<math>\{ \text{WOT-open sets in } B(X,Y)\} \subseteq \{\text{SOT-open sets in }B(X,Y)\} \subseteq \{\text{operator-norm-open sets in }B(X,Y)\},</math>

and these inclusions may or may not be strict depending on the choices of <math>X</math> and <math>Y</math>.

The WOT on <math>B(X,Y)</math> is a formally weaker topology than the SOT, but they nevertheless share some important properties.  For example,

:<math>(B(X,Y),\text{SOT})^*=(B(X,Y),\text{WOT})^*.</math>

Consequently, if <math>S \subseteq B(X,Y)</math> is convex then

:<math>\overline{S}^\text{SOT}=\overline{S}^\text{WOT},</math>

in other words, SOT-closure and WOT-closure coincide for convex sets.