The Separation Theorems
The Separation Theorems
Theorem 1 (The Separation Theorem): Let $X$ be a locally convex topological vector space and let $K_1$ and $K_2$ be disjoint closed convex subsets of $X$. If $K_1$ is compact then there exists a continuous linear functional $f$ on $X$ and and a constant $c \in \mathbb{R}$ such that $f(k_1) < c < f(k_2)$ for all $k_1 \in K_1$ and for all $k_2 \in K_2$. |
Corollary 2 (The Hyperplane Separation Theorem): Let $X$ be a locally convex topological vector space. If $K$ is nonempty, closed, and convex, and if $x_0 \in X \setminus K$ then there exists a continuous linear functional $f$ on $X$ such that $\displaystyle{f(x_0) < \inf_{k \in K} f(k)}$. |
Corollary 3: Let $X$ be a normed linear space. If $K$ is a nonempty, norm closed, and convex subset of $X$ and if $x_0 \in X \setminus K$ then there exists a bounded linear functional $f \in X^*$ such that $\displaystyle{f(x_0) < \inf_{k \in K} f(k)}$. |
Note that Corollary 3 follows immediately from corollary 2 since every normed linear space is a locally convex topological vector space.
Corollary 4: Let $X$ be a normed linear space. If $K$ is a nonempty, norm closed, and convex subset of $X$ and if $x_0 \in X \setminus K$ then there exists a bounded linear functional $f \in X^*$ with $\| f \| = 1$ such that $\displaystyle{f(x_0) < \inf_{k \in K} f(k)}$. |
- Proof: By corollary 3 there is a bounded linear functional $g \in X^*$ such that:
\begin{align} \quad g(x_0) < \inf_{k \in K} g(x) \end{align}
- Note that $g \neq 0$, otherwise the inequality above cannot be strict. Thus, let $f = g/\| g \|$. Then:
\begin{align} \quad f(x_0) = \frac{g(x_0)}{\| g \|} < \frac{1}{\| g \|} \inf_{k \in K} g(x) = \inf_{k \in K} \frac{g(x)}{\| g \|} = \inf_{k \in K} f(x) \end{align}