A Reflexive Linear Space X is Separable IFF X is Separable

A Reflexive Linear Space X is Separable IFF X is Separable

Recall from the If a Normed Linear Space 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. $\blacksquare$
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