Hahn–Banach theorem

Remarks

The proof first extends $f$ to $M + \R x_0$ for one vector $x_0 \notin M$. The value $F(x_0)$ must be a number $c$ with $\sup_{m \in M} (f(m) - p(m - x_0)) \le c \le \inf_{m \in M} (p(m + x_0) - f(m))$, and subadditivity of $p$ shows that such a $c$ exists. Zorn's lemma applied to the partial extensions dominated by $p$ then gives an extension to all of $X$.

The most used case takes $p(x) = \norm{f}\,\norm{x}$ on a real normed vector space. It shows that every bounded linear functional on a subspace extends to the whole space with the same operator norm. The complex case of this norm version follows by applying the real theorem to the real part of $f$. A consequence is that for each $x \ne 0$ in a normed space there is a functional $F$ of norm $1$ with $F(x) = \norm{x}$, so the continuous dual space separates points.

Going deeper

For a student

In a course on functional analysis this is the first of the main theorems, next to the uniform boundedness principle and the bounded inverse theorem. Its norm form gives the isometric embedding of a space into its bidual. Its geometric form is the Hahn–Banach separation theorem, which separates disjoint convex sets by a closed hyperplane. The two forms are linked by the Minkowski functional of a convex neighbourhood of $0$. In a general real vector space the proof needs Zorn's lemma or a weaker choice principle.