Normed Dual Space is Banach Space

Theorem

Let $\GF \in \set {\R, \C}$.

Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space over $\GF$.

Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual of $\struct {X, \norm {\, \cdot \,}_X}$.

Then $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ is a Banach space.

Proof

By definition, we have:

$X^\ast = \map B {X, \GF}$

and:

$\norm {\, \cdot \,}_{X^\ast} = \norm {\, \cdot \,}_{\map B {X, \GF} }$

From Real Number Line is Banach Space and Complex Plane is Banach Space, $\GF$ is a Banach space.

So from Space of Bounded Linear Transformations is Banach Space, $\struct {X, \norm {\, \cdot \,}_X}$ is a Banach space.

$\blacksquare$