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? 



If the cardinality of $B(X)$, the space of operators on $X$, is continuum, must $X$ be separable?
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](https://en.wikipedia.org/wiki/Split_interval) $K$. 


The $w^\*$-separability of the dual space $X^\*$ was proved by Corson, see Theorem 12.43 in [this book](https://www.amazon.co.uk/Functional-Analysis-Infinite-Dimensional-Geometry-Mathematics/dp/1441929126/ref=sr_1_2/257-8911836-5630525?s=books&ie=UTF8&qid=1528542703&sr=1-2). The equality $|X^\*|=\mathfrak c$ can be seen analyzing the structure of (probability) measures on the compact space $K$.


