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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 $L^1$).

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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.

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