\begin{definition}[Definition:Second Normed Dual]
Let $\struct {X, \norm \cdot_X}$ be a normed vector space.
Let $\struct {X^\ast, \norm \cdot_{X^\ast} }$ be the normed dual of $\struct {X, \norm \cdot_X}$.
We define the '''second normed dual''', written $\struct {X^{\ast \ast}, \norm \cdot_{X^{\ast \ast} } }$ as the normed dual of $\struct {X^\ast, \norm \cdot_{X^\ast} }$.
\end{definition}