On complex strictly convex complexifications of Banach spaces

We show that every real separable normed space may be complexified to a complex strictly convex normed space. The same result is obtained also for some classes of nonseparable spases, for example, for spases $X$ with 1-norming separable subspases in $X^*$; however, a space $\ell_\infty(\Gamma)$ has no complex strictly convex complexifications.