Reflexive Normed Linear Spaces

Reflexive Normed Linear Spaces

Definition: Let $X$ be a normed linear space. Then $X$ is said to be Reflexive if $J(X) = X^{**}$.

Observe that a normed linear space $X$ being reflexive is equivalent to saying that the canonical embedding $J : X \to X^{**}$ defined for all $x \in X$ by $J(x) = J_x$ is surjective.

The following theorem tells us exactly when a normed linear space $X$ is reflexive.

Theorem 1: Let $X$ be a normed linear space. Then $X$ is reflexive if and only if the weak* topology on $X^*$ is the weak topology on $X^*$.
  • Proof: $\Rightarrow$ Suppose that $X$ is reflexive. Then:
(1)
\begin{align} \quad J(X) = X^{**} \quad (\dagger) \end{align}
  • Now by definition, the weak* topology on $X^*$ is the $J(X)$-weak topology on $X^*$.
  • Also by definition, the weak topology on $X^*$ is the $(X^*)^* = X^{**}$-weak topology on $X^*$.
  • By $(\dagger)$ these two topologies on $X^*$ are the same.
  • $\Leftarrow$ Let $\Omega \in X^{**}$. Then $\Omega : X^* \to \mathbb{C}$ is continuous with respect to the weak topology on $X^*$. But then $\Omega$ is continuous with respect to the weak* topology on $X^*$, i.e., $\Omega$ is continuous with respect to the $J(X)$-weak topology on $X^*$. By the theorem on the The W-Weak Topology on a Normed Linear Space page, we must have that:
(2)
\begin{align} \quad \Omega \in J(X) \end{align}
  • Therefore:
(3)
\begin{align} \quad X^{**} \subseteq J(X) \end{align}
  • And since $J : X \to X^{**}$ we also have that $X^{**} \supseteq J(X)$. Therefore:
(4)
\begin{align} \quad X^{**} = J(X) \end{align}
  • So $X$ is reflexive. $\blacksquare$
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