The Hahn-Banach theorem is a foundational result in functional analysis. Roughly speaking, it asserts the existence of a great variety of bounded (and hence continuous) linear functionals on an normed vector space, even if that space happens to be infinite-dimensional. We first consider an abstract version of this theorem, and then give the more classical result as a corollary.

Let $V$ be a real, or a complex vector space, with $K$ denoting the corresponding field of scalars, and let $$\pnorm:V\rightarrow\reals^+$$ be a seminorm on $V$ .

Theorem 1   Let $f:U\to K$ be a linear functional defined on a subspace $U\subset V$ . If the restricted functional satisfies $$\vert f(\bu)\vert\leq \snorm{\bu},\quad \bu\in U,$$ then it can be extended to all of $V$ without violating the above property. To be more precise, there exists a linear functional $F:V\to K$ such that

Definition 2   We say that a linear functional $f:V\to K$ is bounded if there exists a bound $B\in\reals^+$ such that \begin{equation} \label{eq:bdef} \vert f(\bu)\vert \leq B \snorm{\bu},\quad \bu\in V. \end{equation}If $f$ is a bounded linear functional, we define $\Vert f\Vert$ , the norm of $f$ , according to $$\Vert f \Vert = \sup \{ \vert f(\bu)\vert : \snorm{\bu} = 1 \}.$$ One can show that $\Vert f\Vert$ is the infimum of all the possible $B$ that satisfy ()