\begin{definition}[Definition:Normed Dual Space]
Let $\struct {X, \norm \cdot_X}$ be a normed vector space.
Let $X^\ast$ be the vector space of bounded linear functionals on $X$. 
Let $\norm \cdot_{X^\ast}$ be the norm on bounded linear functionals.
We say that $\struct {X^\ast, \norm \cdot_{X^\ast} }$ is the '''normed dual space''' of $X$.
\end{definition}