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.



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