Hahn-Banach theorem

The Hahn-Banach theorem is a foundational result in functionalanalysis. Roughly speaking, it asserts the existence of a greatvariety of bounded (and hence continuous) linear functionals on annormed vector space, even if that space happens to beinfinite-dimensional. We first consider anabstract version of this theorem, and then give the more classicalresult as a corollary.

Let $V$ be a real, or a complex vector space, with $K$denoting the corresponding field of scalars, and let

$$\mathrm{p}:V\to {ℝ}^{+}$$

be a seminorm on $V$.

Theorem 1

Let $f\mathrm{:}U\mathrm{\to }K$ be a linear functional defined on a subspace$U\mathrm{\subset }V$. If the restricted functional satisfies

$$|f(𝐮)|\le \mathrm{p}(𝐮),𝐮\in U,$$

then it can be extended to all of $V$ without violating the aboveproperty. To be more precise, there exists a linear functional$F\mathrm{:}V\mathrm{\to }K$ such that

$F(𝐮)$$=f(𝐮),𝐮\in U$

$|F(𝐮)|$$\le \mathrm{p}(𝐮),𝐮\in V.$

Definition 2

We say that a linear functional $f\mathrm{:}V\mathrm{\to }K$ is bounded ifthere exists a bound $B\mathrm{\in }{\mathrm{R}}^{\mathrm{+}}$ such that

$$|f(𝐮)|\le B\mathrm{p}(𝐮),𝐮\in V.$$(1)

If $f$ is a bounded linear functional, we define $\mathrm{\parallel }f\mathrm{\parallel }$, thenorm of $f$, according to

$$\parallel f\parallel =sup\{|f(𝐮)|:\mathrm{p}(𝐮)=1\}.$$

One can show that $\mathrm{\parallel }f\mathrm{\parallel }$ is the infimum of all the possible$B$ that satisfy (1)

Theorem 3 (Hahn-Banach)

Let $f\mathrm{:}U\mathrm{\to }K$ be a bounded linear functional defined on a subspace$U\mathrm{\subset }V$. Let ${\mathrm{\parallel }f\mathrm{\parallel }}_U$ denote the norm of $f$ relativeto the restricted seminorm on $U$. Then there exists a boundedextension $F\mathrm{:}V\mathrm{\to }K$ with the same norm, i.e.

$${\parallel F\parallel }_V={\parallel f\parallel }_U.$$