Completeness and separability of $B(X)$ (bounded linear operators on $X$)

Question

Assume $X$ is a separable Banach space with norm $|| \cdot||$. Consider $\{f_n\}_{n \in N}$ a countable dense subset of $X$ and equip $B(X)$ (bounded linear operators on $X$) with the following metric: \begin{align} & d(A,B)=\sum_{n=0}^\infty 2^{-n} (1+||f_n||)^{-1} || A f_n-B f_n || \end{align}

Does this metric make $B(X)$ complete and separable?

PS: completeness seems quite easy to prove. The interesting part is separability.

PS2: below it is shown that $B_1(X)$ (the unit ball of $B(X)$ in the operator norm metric) is separable. What about the whole $B(X)$?

http://thales.doa.fmph.uniba.sk/sleziak/texty/rozne/pozn/books/kechrisexercises.pdf

Answer 1

The metric describes pointwise convergence on the dense set $\lbrace f_n:n\in\mathbb N\rbrace$ which is a vector space topology (even locally convex) which is clearly coarser than the usual operator norm topology. Therefore, the metric cannot be complete because otherwise the open mapping theorem (for Frechet spaces) implies that the metric and the norm give the same topology which is clearly not the case if $X$ is infinite dimensional.

Just an idea concerning separability: If the dual $X'$ is separable one could try to show that $X' \otimes X = \lbrace$finite rank operators$\rbrace$ is dense which would yield separability.