If an operator $T: A\rightarrow B$ satisfying for every sequence $\{X_n\}$ weakly converging to $X$, we have $TX_n \rightarrow TX$ in weak topology. Then, is $T$ weak-weak continuous?
And in the WOT/SOT, does WOT-WOT(SOT-SOT) continuity of $T$ mean: for every sequence $\{X_n \in B(H)\}$ weakly converging to $X$ in WOT, then $TX_n \rightarrow_{WOT} TX$.
And, if an element $X$ in $\overline{V}^{WO}$or $\overline{V}^{SO}$, can we say there is a sequence $\{X_n\in V\}$ converging to $X$ in WOT/SOT? What about in the weak topology?
I am not clear about sequential continuity/closure and continuity/closure.
I know in norm space sequential continuity and continuity is equivalent, so is the sequential closure and closure.