# Hahn-Banach theorem

Originally published at 狗和留美者不得入内. You can comment here or there.

I’m pleased to say that I find the derivation of the Hahn-Banach theorem pretty straightforward by now. Let me first state it, for the real case.

Hahn-Banach theorem: Let $V$ be a real vector space. Let $p: V \to \mathbb{R}$ be sublinear. If $f : U \to \mathbb{R}$ be a linear functional on the subspace $U \subset V$ with $f(x) \leq p(x)$ for $x \in U$, then there exists a linear extension of $f$ to all of $V$ (call it $g$) such that $f(x) \leq g(x)$ for $x \in V$ with $f(x) = g(x)$ for $x \in U$ and $g(x) \leq p(x)$ for all $x \in V$.

To show this, start by taking any $x_0 \in V \setminus U$. We wish to assign some $\alpha$ to $x_0$ that keeps $p$ as the dominating function in the vector space $U + \mathbb{R}x_0$. For this to happen, applying the linearity of $f$ and the domination constraint, we can derive

$\frac{f(y) - p(y - \lambda x_0)}{\lambda} \leq \alpha \leq \frac{p(y+\lambda x_0) - f(y)}{\lambda}, \quad y \in U, \lambda > 0$.

This reduces to

$\sup_{y \in U} p(y+x_0) - f(y) \leq \inf_{y \in U} f(y) - p(y-x_0)$.

Such can be proven via

$f(y_1) + f(y_2) = f(y_1 + y_2) \leq p(y_1 + y_2) \leq p(y_1 - x_0) +p(y_2 + x_0), \quad y_1, y_2 \in U$.

Now take the space of linear functionals defined on some specific subspace dominated by $p$. Denote an element of it as $(f, U)$. We introduce a partial order wherein $(f, U) \leq (f', U')$ iff $f(x) = f'(x)$ for $x \in U$ and $U \subset U'$. We can apply Zorn’s lemma on this, as we can take the union to derive an upper bound for any chain. Any maximal element is necessarily $(g, V)$ as if the domain is not the entire vector space, we can by above construct a larger element.

• #### My thoughts on Deng Xiaoping and the Cultural Revolution

Originally published at 狗和留美者不得入内. You can comment here or there. I’ll say that I‘ve talked with people in the older generation who think…

• #### Rebuttal to a comment on this matter of “market vs command economy” regarding PRC

Originally published at 狗和留美者不得入内. You can comment here or there. Original comment on Steve Hsu’s blog: http://disq.us/p/2f9gtns The…

• #### More on America’s disingenuousness and its overrated achievements and capabilities

Originally published at 狗和留美者不得入内. You can comment here or there. http://disq.us/p/2f75cui I generally agree with the thesis that America is in…

• Post a new comment

#### Error

Anonymous comments are disabled in this journal

default userpic