Let $X$ be a Banach space and $B(X)$ be the space of bounded operators on $X$. 

Assume the strong operator topology on $B(X)$ is separable and  cardinal number of $B(X)$ is of the continuum.  

>**Question**. Can we conclude the norm topology on $X$ is separable?

**Remark.** It was proved by [Tomek Kania][1] that the assumption $|B(X)|=\mathfrak{c}$ is not solely  enough to get the separability assertion.   


  [1]: https://mathoverflow.net/questions/302286/if-the-cardinality-of-bx-the-space-of-operators-on-x-is-continuum-must