Kakutani's Theorem

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Theorem

Let $X$ be a normed vector space.

Let $w$ be the weak topology on $X$.

Let $B_X^-$ be the closed unit ball in $X$.

Let $X^{\ast \ast}$ be the second normed dual of $X$.

Then $X$ is reflexive if and only if $\struct {B_X^-, w}$ is compact.

Proof

Let $B_{X^{\ast \ast} }^-$ be the closed unit ball in $X^{\ast \ast}$.

Let $w^\ast$ be the weak-* topology on $X^{\ast \ast}$.

Let $\iota : X \to X^{\ast \ast}$ be the evaluation linear transformation.

Necessary Condition

Suppose that $X$ is reflexive.

By Normed Vector Space is Reflexive iff Closed Unit Ball in Original Space is Mapped to Closed Unit Ball in Second Dual, $\iota B_X^- = B_{X^{\ast \ast} }^-$.

From Evaluation Linear Transformation on Normed Vector Space is Weak to Weak-* Homeomorphism onto Image, we have that:

$\iota : \struct {X, w} \to \struct {\iota X, w^\ast}$ is a homeomorphism.

From Restriction of Homeomorphism is Homeomorphism, we have that:

$\iota : \struct {B_X^-, w} \to \struct {\iota B_X^-, w^\ast} = \struct {B_{X^{\ast \ast} }^-, w^\ast}$ is a homeomorphism.

From the Banach-Alaoglu Theorem, $\struct {B_{X^{\ast \ast} }^-, w^\ast}$ is compact.

Since $\struct {B_X^-, w}$ is homeomorphic to $\struct {B_{X^{\ast \ast} }^-, w^\ast}$, we have that $\struct {B_X^-, w}$ is compact.

$\Box$

Sufficient Condition

Suppose that $\struct {B_X^-, w}$ is compact.

From Evaluation Linear Transformation on Normed Vector Space is Weak to Weak-* Continuous Embedding into Second Normed Dual, $\iota : \struct {X, w} \to \struct {X^{\ast \ast}, w^\ast}$ is continuous.

From Continuous Image of Compact Space is Compact, $\iota B_X^-$ is a compact subset of $\struct {X^{\ast \ast}, w^\ast}$.

In particular, $\iota B_X^-$ is closed in $\struct {X^{\ast \ast}, w^\ast}$.

So:

$\map {\cl_{w^\ast} } {\iota B_X^-} = \iota B_X^-$

from Set is Closed iff Equals Topological Closure.

From Goldstine's Theorem, we have that:

$\map {\cl_{w^\ast} } {\iota B_X^-} = B_{X^{\ast \ast} }^-$

So, we have:

$\iota B_X^- = B_{X^{\ast \ast} }^-$

From Normed Vector Space is Reflexive iff Closed Unit Ball in Original Space is Mapped to Closed Unit Ball in Second Dual, $X$ is reflexive.

$\blacksquare$

Source of Name

This entry was named for Shizuo Kakutani.