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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.

A normed vector space over $\mathbb{R}$ is a real vector space $X$ with a function $\Vert \cdot \Vert :X\to [0,\infty )$ satisfyingIts distance is $d(x,y)=\Vert x-y\Vert$. It is a Banach space if this metric is complete: every Cauchy sequence converges to an element of $X$.