# 泛函中的重要定理之 Hahn-Banach 定理

Hahn-Banach Theorem is one of the core theorems of linear functional analysis. The Hahn-Banach Theorem in Vector Space is also called Analytic Form of Hahn-Banach Theorem. Two corollaries are especially important: Hahn-Banach Theorem in Normed Vector Space, and Geometric Form of Hahn-Banach Theorem.

## Analytic Form

Let $X$ be a real vector space, and $p$ is a sublinear functional in $X$, that is, $p:X\rightarrow\mathbb{R}$ is a function satisfies the following properties:

$$\begin{eqnarray}p(\alpha x) = \alpha p(x),\quad \forall \alpha>0 \text{ and } x\in X,\\p(x+y) \leqslant p(x)+p(y)\quad \forall x,y\in X.\end{eqnarray}$$
And let $Y$ be a subspace of $X$, $l:Y\rightarrow\mathbb{R}$ is a linear functional in $Y$ which satisfies
$$l(y)\leqslant q(y),\quad \forall y\in Y.$$
Then there exists a linear functional $\widetilde{j}:X\rightarrow\mathbb{R}$, such that
$$\widetilde{l}(y)=l(y),\quad \forall y\in Y.\quad \text{ and } \widetilde{l}(y)\leqslant p(x),\quad \forall x\in X.$$