A Banach space is reflexive if and only if its dual is reflexive

Solution 1:

I'm assuming that you have two Theorems at your disposal, easily proven:

Theorem 1: If a Banach space $X$ is reflexive then its dual space $X'$ is reflexive.

Theorem 2: A closed subspace of a reflexive Banach space is reflexive.

Claim: Let $X$ be a Banach space. If $X'$ is reflexive then $X$ is reflexive.

Proof: Suppose $X'$ is reflexive. By Theorem 1 it follows that $X''$ is reflexive. If we consider the Canonical mapping $J : X \to X''$ it follows that $J(X)$ is a subspace of $X''$. Since $X \cong J(X)$ and $X$ is a Banach space, then $J(X)$ is a Banach space and hence closed. By Theorem 2 we can conclude that $J(X)$ is reflexive. But since $X \cong J(X)$ conclude that $X$ is reflexive.

Moreover a consequence of this is that a Banach space is reflexive if and only if its dual space is reflexive.

Solution 2:

It can be shown directly:

Let $J : X \to X^{**}$ and $J_*: X^* \to X^{***}$ be the canonical injections. Suppose by contradiction that $JX \subsetneq X^{**}$; using Hahn-Banach theorem, there exists $\zeta \in X^{***}$ such that $\zeta \neq 0$ and $\zeta \equiv 0$ on $JX$.

Because $X^*$ is reflexive, there exists $\theta \in X^*$ such that $\zeta = J_*\theta$. For all $x \in X$:

$$0= \langle \zeta,Jx \rangle= \langle J_*\theta,Jx \rangle = \langle Jx,\theta \rangle= \langle \theta,x \rangle$$

You deduce that $\theta=0$ and therefore $\zeta=0$: a contradiction.

Solution 3:

Really a Banach space $X$ is reflexive if and only if $X'$ is reflexive.

$$X\textrm{ is reflexive}\Longrightarrow X'\textrm{ is reflexive.}\tag{1}$$

Proof. By Banach-Alaoglu-Bourbaki theorem the closed ball $B_{X'}$ is closed w.r.t. the weak-* topology $\sigma(X',X)$. By the reflexivity of $X$ we have $\sigma(X',X'')=\sigma(X',X).$ So $B_{X'}$ is closed w.r.t. the weak topology $\sigma(X',X),$ that is $X'$ is reflexive.$\square$

$$X'\textrm{ is reflexive}\Longrightarrow X\textrm{ is reflexive.}\tag{2}$$

Proof. By hypothesis and by (1) we get that $X''$ is reflexive, and therefore even its closed vector subspace $J(X)$ is reflexive. But the canonical injection $J:X\to X''$ is an isometry so $X$ is reflexive.$\square$