Functional Analysis 6 -- Hahn-Banach Theorem Separation

This is course note from Siran Li-Shanghai Jiao Tong University.

Definition: Locally Convex

A topological vector space $(X, T)$ is locally convex iff $\exists$ local base $B$ at $0$, s.t. every member of $B$ is convex.

Definition: Locally Bounded

$(X, T)$ being a TVS is locally bounded iff $\exists$ local base $B$ at $0$, s.t. $0 \in X$ has bounded neighborhood.

Fact:(Rudin) Let $X$ be a TVS. 1. $X$ is metrisable $\Leftrightarrow$ $X$ has a countable local base. 2. $X$ is normable $\Leftrightarrow$ $X$ is a locally convex and local bounded.

Definition: Absorbing

$\mathcal{A} \subset X$ is absorbing iff $\forall x \in X, \exists t \ge 0, s.t. \ x \in t\mathcal{A}.$

Definition: Minkowski Functional

Let $X$ be a TVS, $\mathcal{A} \subset X.$ The Minkowski functional of $\mathcal{A}$ is $\mu_{\mathcal{A}} : X \rightarrow [0, \infty],$ given by $\mu_{\mathcal{A}} := \inf\{ t>0 : x \in t\mathcal{A}\}.$

Remark: 1. If $\mathcal{A}$ is absorbing, then $\mathcal{A}: X \rightarrow [0, \infty[.$ 2. For any TVS, any neighborhood of $0$ is absorbing.

Definition: Balanced > $B \subset X$ is balanced iff $\alpha B \subset B, \forall \alpha \in \mathbb{K}$ with $|\alpha| \le 1.$

An intuitive understand: balanced + convex + absorbing $\sim$ ball.

Theorem 6.1

Let $X$ be a VS. Let $p: X \rightarrow \mathbb{R}_{+}$ be a seminorm. Then * $p(0) = 0$ * $|p(x) - p(y)| \le p(x-y)$ * $p \ge 0$ * $\{p = 0\}$ is a subspace. * $B = \{ p < 1\}$ is convex, balanced, absorbing neighborhood of $0$ with $p = \mu_{B}.$

proof:[[TODO]]

Theorem 6.1 told us that starts from a seminorm, we can get a Minkowski functional with a balanced, absorbing, convex set.

Theorem 6.2

Let $X$ be a VS and let $A \subset X$ be convex and absorbing. Then $\mu_A$ is a sublinear functional. If in addition, $A$ is balanced, then $\mu_A$ is a seminorm. Moreover, $B = \{\mu_A < 1\} \subset A \subset C=\{\mu_A \le 1\}$ and $\mu_B = \mu_A = \mu_C.$

proof:[[TODO]]

Theorem 6.2 told us that starts from a subset, what is the Minkowski functional.

Example: Let $X = \mathbb{R}^2, p(x) = dist(x, \mathbb{R}^2 \times \{0\}),$ then $\mu_B = p.$

Now, consider $A \subset X$ to be a convex, balanced, absorbing subset. 1. By theorem 2 we have that $\mu_A$ is a seminorm. 2. By theorem 1 we have that $|\mu_A(x) - \mu_A(y)| \le \mu_A(x-y) \le r$ whenever $\frac{x-y}{r} \in A.$ By 1., 2. we have $\mu_A$ is continuous.

Theorem 6.3

Let $X$ be TVS with a convex, balanced local base $B$. Then 1. $V = \{\mu_V < 1\}, \forall V \in B/$ 2. $\{\mu_V: V \in B\}$ is a separating family of continuous seminorms on $X.$

Theorem 6.4: Hahn-Banach Theorem Separation (See Bressan pp31-32)

Let $X$ be a NVS over $\mathbb{R}.$ Let $A, B \subset X$ be disjoint convex, non-empty subset. 1. If $A$ is open, then $f X^*, , $ s.t. $f(a) < \lambda \le f(b), \forall a \in A, b \in B.$ 2. If $A$ is compact and $B$ is closed, then $\exists g \in X^*, \lambda_1, \lambda_2 \in \mathbb{R},$ s.t. $g(a) \le \lambda_1 < \lambda_2 \le g(b), \forall a \in A, b \in B.$

proof:[[Proof of Theorem 6.4]]