If X is Reflexive then X is Separable IFF X* is Separable

Recall from the If X* is Separable then X is Separable page that if $X$ is a normed linear space and $X^*$ is separable then $X$ is separable.

When $X$ is further a reflexive normed linear space then we can say more.

Theorem 1: Let $X$ be a normed linear space. If $X$ is reflexive then $X$ is separable if and only if $X^*$ is separable.

• Proof: $\Rightarrow$ Suppose that $X$ is separable. Since the canonical embedding $J : X \to X^{**}$ is continuous, the image $J(X)$ is separable. But since $X$ is reflexive we have that:

(1)

\begin{align} \quad J(X) = X^{**} \end{align}

• Therefore $X^{**}$ is separable. But by the theorem referenced at the top of this page, this implies that $X^*$ is separable.

• $\Leftarrow$ Suppose that $X^*$ is separable. Then by the theorem referenced at the top of this page, $X$ is separable. $\blacksquare$