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How to show that a Banach space $X$ is reflexive if its dual $X'$ is reflexive?

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First show that if $X'$ is reflexive then $X''$ is reflexive. Then what can you say about $\hat X$? –  matt Jun 1 '12 at 6:26
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@matt: what's $\hat{X}$? –  t.b. Jun 1 '12 at 6:27
    
@t.b. $\hat X$ is canonical embedding of $X$ in $X''$. Sorry I should mentioned this. –  matt Jun 1 '12 at 6:30

3 Answers 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.$\square$

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Appealing to Alaoglu is a bit of an overkill: if $X$ is reflexive then the canonical map $\iota_{X}\colon X \to X^{\ast\ast}$ is an isomorphism. Hence $\iota_{X^\ast} = ((\iota_{X})^{\ast})^{-1} = ((\iota_{X})^{-1})^\ast$ is an isomorphism. –  t.b. Jun 1 '12 at 7:00
    
Dear Theo, I haven't thought this argument, it is simpler than the one I have wrote. Thank you. –  Giuseppe Tortorella Jun 1 '12 at 7:08
    
@t.b. could you expand that thought? I don't understand your argument. –  Parakee Nov 9 '12 at 3:21
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@Parakee: 1. Let $\iota_X\colon X \to X^{\ast\ast}$ and $\iota_{X^\ast}\colon X^\ast \to X^{\ast\ast\ast}$ be the canonical inclusions. Then we have that $(\iota_X)^\ast \circ (\iota_{X^\ast}) = \operatorname{id}_{X^\ast}$ by a direct verification. 2. If an operator is invertible then so is its adjoint $(\iota_X)^\ast$. 3. If an operator $S$ is right inverse to an invertible operator $T$ then $S$ is itself invertible and equal to the inverse: $S = T^{-1}$. 4. Apply this to $T = (\iota_X)^\ast$ and $S = \iota_{X^\ast}$. –  t.b. Nov 29 '12 at 14:41

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.

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It can be shown directly:

Let $J : X \to X^{**}$ and $J_*: X^* \to X^{***}$ be the canonical surjections. 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 reflexif, 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.

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Seirios, $\zeta \in X^{***}$ and $Jx \in X^{**}$, so how exactly can an inner product $\langle \zeta ,Jx \rangle $ be defined? Secondly, there is no assumption of inner products defined on $X, X^{**}$ , etc. $X$ is merely a Banach space by hypothesis. Indeed, if an inner product were defined on $X$, then we would immediately have that $X$ is reflexive, as every Hilbert space is reflexive. –  Marc Nye Oct 4 '13 at 1:18
    
$\langle \cdot, \cdot \rangle$ is not an inner product, but a duality bracket, that is: $\langle \zeta,x \rangle$ is the evaluation of $\zeta \in X^*$ at $x \in X$. Isn't a usual notation? –  Seirios Oct 4 '13 at 7:17

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