Reflexive Normed Linear Spaces

Recall from The Natural Embedding J page that if $(X, \| \cdot \|_X)$ is a normed linear space then the natural embedding from $X$ to $X^{**}$ is the map $J : X \to X^{**}$ defined for all $x \in X$ by:

(1)

\begin{align} \quad J(x) = \hat{x} \end{align}

Where $\hat{x} : X^* \to \mathbb{R}$ defined for all $f \in X^*$ by $\hat{x}(f) = f(x)$. We proved the following about $J$:

• 1. $J$ is a bounded linear operator from $X$ to $X^{**}$.

• 2. $\| J(x) \| = \| x \|_X$ for all $x \in X$, that is, $J$ is an isometry.

• 3. $J$ is injective (since it is an isometry).

We now define what it means for a normed linear space to be reflexive.

Definition: Let $(X, \| \cdot \|_X)$ be a normed linear space and let $J : X \to X^{**}$ be the natural embedding of $X$ into $X^{**}$. $X$ is said to be Reflexive if $J(X) = X^{**}$, that is, $J$ is surjective.