I am reading a book about C*-algebra. In the book,
Let $\phi$ be a linear functional on $B(H)$ ($H$ denotes a Hilbert space), if $\phi$ is strongly continuous, therefore, there exist vectors $\xi_{1}, \xi_{2},...,\xi_{n}$ in $H$ and $\delta>0$ such that $|\phi(a)|\leq1$, whenever $max_{k}\{||a(\xi_{k})||\}\leq\delta$ for all $a\in B(H)$.
I can not understand this conclusion, could someone explain to me?
Sets of the form $V(a_0,\xi_1,\xi_2,\dotsc,\xi_n,\delta) = \{ a\in B(H) \colon \|(a-a_0)(\xi_i)\| < \delta \text{ for } i=1,2,\dotsc,n \}$ for $\delta>0$, $n\in\mathbb{N}$, $\xi_i\in H$, $a_0\in B(H)$ forms a basis in the strong operator topology. Let $U$ be the open unit disk in $\mathbb{C}$. Since $\phi$ is continuous from strong-operator topology of $B(H)$ to $\mathbb{C}$ and $\phi(0)=0$, $\phi^{-1}(U)$ is open and so there exists $\xi_i\in H$ and $\delta>0$ so that $\phi(V(\xi_1,\xi_2,\dotsc,\xi_n,\delta)) \subseteq U$.