TITLE: If the cardinality of $B(X)$, the space of operators on $X$, is continuum, must $X$ be separable?
QUESTION [8 upvotes]: Does there exits any non-separable Banach space $X$ such that the size (cardinal number) of $B(X)$, bdd linear operators on $X$, is just of the continuum?

REPLY [6 votes]: An example of a non-separable Banach space $X$ with $|B(X)|=\mathfrak c$ is  any non-separable Banach space $X$ whose dual $X^*$ is $w^*$-separable and has cardinality $|X^*|=\mathfrak c$. 
This follows from the observation that the map $B(X)\to B(X^*)$, $T\mapsto T^*$, is injective and hence for a countable $w^*$-dense set $D$ in $X^*$ we have  $$|B(X)|\le |B(X^*)|\le |(X^*)^{D}|=|X^*|^\omega=\mathfrak c^\omega=\mathfrak c.$$
A ZFC-example of a non-separable Banach space $X$ whose dual space $X^*$ is $w^*$-separable and has cardinality $|X^*|=\mathfrak c$ is the Banach space $X=C(K)$ of continuous functions on the Alexandroff two-arrow space $K$. 
The $w^*$-separability of the dual space $X^*$ was proved by Corson, see Theorem 12.43 in this book. The equality $|X^*|=\mathfrak c$ can be seen analyzing the structure of (probability) measures on the compact space $K$.