Reflexive but not separable space

Question

I'm trying to find an example of normed vector space that is reflexive but not separable.

(Separable but not reflexive is easy, for example $L^1$).

Answer 1

Let $X$ be uncountable and $\mu$ the counting measure on $X$. Then for all $p \in (1, \infty)$, $L^p(X, \mu)$ is reflexive but not separable.