Strongly-Continuous linear functionals on $\mathcal{B}(H)$

Question

Suppose $H$ is a complex Hilbert space and $$w: \mathcal{B}(H) \longrightarrow \mathbb{C}$$ is a bounded linear functional on $\mathcal{B}(H)$ such that $w$ is continuous even if $\mathcal{B}(H)$ is given the strong operator topology. I am supposed to prove that there exists $c>0$ and $h_1, \dots, h_n \in H$ such that for all $x \in \mathcal{B}(H)$, one has $$|w(x)| \leq c \sqrt{\sum_{i = 1}^n \|x(h_i)\|^2}.$$ I have no idea how to begin. To me, the strong operator topology is given by the family of seminorms $$\{\|x\|_h = \|x(h)\| : h \in H\}.$$

Answer 1

Hint: What do the open sets of the strong operator topology look like (specifically, what is a basis for this topology)? What does this say about $\{x:|w(x)|<1\}$?