 Solution

The continuous dual space $X^{\prime }$ is the vector space of bounded scalar-valued linear functionals, with operator normThis makes it a normed space, and in fact a Banach space whether or not $X$ is complete.

The norm-preserving Hahn-Banach theorem states that a bounded linear functional $f_0$ on any linear subspace $M\subseteq X$ extends to a bounded linear functional $f$ on all of $X$ with $f{|}_M=f_0$ and $\Vert f\Vert =\Vert f_0\Vert$. The subspace need not be closed. This holds over both the real and complex scalar fields, using the complex version when necessary.