Let $C[0,1]$ be the space of all Real valued continuous functions on $[0,1]$ with the usual supremum norm. Does there exist an equivalent renorming on $C[0,1]$ such that the corresponding dual norm is strictly convex?



Renorming of $C[0,1]$ for a strictly convex dual
One typically equivalently renorms a Banach space $Y$ to be strictly convex by finding an injective operator $S$ from $Y$ into some strictly convex space $Z$ and defining the new norm on $Y$ by $\|y\| +\|Sy\|$. When $Y$ and $Z$ are dual spaces and $S$ is weak$^\*$ to weak$^\*$ continuous, the new norm is a dual norm (the new unit ball of $X^\*$ is weak$^\*$ closed because $T^\*$ is weak$^\*$ continuous).


So let $X$ be any separable space and take an operator $T:\ell^2 \to X$ that has dense range. Renorm $X^\*$ by
$!F! := \| F \|\_{X^\*} + \|T^\*F\|\_2$.


