Image of Evaluation Linear Transformation on Banach Space is Closed Linear Subspace of Second Dual

Theorem

Let $\struct {X, \norm \cdot_X}$ be a Banach space.

Let $\struct {X^{\ast \ast}, \norm \cdot_{X^{\ast \ast} } }$ be the second normed dual of $\struct {X, \norm \cdot_X}$.

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

Then $\map J X$ is a closed linear subspace of $X^{\ast \ast}$.

Proof

From Image of Vector Subspace under Linear Transformation is Vector Subspace:

$\map J X$ is a vector subspace of $X^{\ast \ast}$.

From Image of Evaluation Linear Transformation on Banach Space is Closed:

$\map J X$ is closed in $X^{\ast \ast}$.

$\blacksquare$