The question is: what is the closure of the set of unitary operators on $\mathcal{l}^2$ in the space of bounded operators with the topology generated by all seminorms of the form $p_{x, y}(A) = |(Ax, y)|$, where $x$, $y$ are vectors from $ l^2$.
I tried to prove that the set of unitary operators is closed in this topology, but I realized that this is not true, because I used the fact that every limit point has a sequence of points converging to it in this topology. The only answer i found is that closure is a whole unit ball. So i want to ask you, is this actually true and how to proove it, or the answer is different (and how to proove it in this case)?