Editing Weak operator topology (section)

==Other properties==

The adjoint operation ''T'' → ''T*'', as an immediate consequence of its definition, is continuous in WOT.

Multiplication is not jointly continuous in WOT: again let <math>T</math> be the unilateral shift. Appealing to Cauchy-Schwarz, one has that both ''T<sup>n</sup>'' and ''T*<sup>n</sup>'' converges to 0 in WOT. But ''T*<sup>n</sup>T<sup>n</sup>'' is the identity operator for all <math>n</math>. (Because WOT coincides with the σ-weak topology on bounded sets, multiplication is not jointly continuous in the σ-weak topology.)

However, a weaker claim can be made: multiplication is separately continuous in WOT. If a net ''T<sub>i</sub>'' → ''T'' in WOT, then ''ST<sub>i</sub>'' → ''ST'' and ''T<sub>i</sub>S'' → ''TS'' in WOT.