Normed Vector Space is Reflexive iff Closed Unit Ball in Original Space is Mapped to Closed Unit Ball in Second Dual

Theorem

Let $X$ be a normed vector space.

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

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

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

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

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

Then $X$ is reflexive if and only if $\iota B_X^- = B_{X^{\ast \ast} }^-$.

Proof

Necessary Condition

Suppose that $X$ is reflexive.

Then $\iota : X \to X^{\ast \ast}$ is an isometric isomorphism.

So from Injective Linear Transformation between Normed Vector Spaces sends Closed Unit Ball to Closed Unit Ball iff Isometric Isomorphism, we have that $\iota B_X^- = B_{X^{\ast \ast} }^-$.

$\Box$

Sufficient Condition

Conversely suppose that $\iota B_X^- = B_{X^{\ast \ast} }^-$.

From Linear Isometry is Injective, $\iota$ is injective.

Then we see that $\iota : X \to X^{\ast \ast}$ is an injective linear transformation with:

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

It follows that $\iota$ is an isometric isomorphism from Injective Linear Transformation between Normed Vector Spaces sends Closed Unit Ball to Closed Unit Ball iff Isometric Isomorphism.

$\blacksquare$