Editing Weak operator topology (section)

===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.