Separation theorems for convex sets (geometric Hahn-Banach)

Suppose $B$ is an open ball in a normed vector space $V$ such that $0 \notin B$. Prove that there exists $\varphi \in V^{\prime }$ such that

for all $f \in B$.

Prove that in a real NLS $X$ every closed ball with center at point 0 is the intersection of some family of sets of the form $\left\{ x \in X \mid f(x) \leq a\right\}$, where $f \in X^{*}, a \in \mathbb {R}$.

Let $U_{1}$ and $U_{2}$ be two convex nonempty disjoint sets of a real Banach space $X$, one of which has nonempty interior. Prove that there exist a linear continuous functional $f$ and a number $c \in \mathbb {R}$ such that the affine hyperplane $X_{f, c}$ separates $U_{1}$ and $U_{2}$.

Let $P$ be the subspace of the real space $C[0,1]$ consisting of all polynomials. Prove that $P_{+}$ and $P_{-}$, the sets consisting of polynomials with positive (negative) leading coefficient, are convex, disjoint, but not separated by any hyperplane.

Let $B_{1}$ and $B_{2}$ be two convex closed nonempty disjoint sets of a real Banach space $X$, one of which is compact. Prove that there exist a linear continuous functional $f$ and a number $c \in \mathbb {R}$ such that the affine hyperplane $X_{f, c}$ strictly separates $B_{1}$ and $B_{2}$.

Let $\mathscr {X}$ be a real TVS. Prove the following.

If $f : \mathscr {X} \to \mathbb {R}$ is a continuous $\mathbb {R}$-linear functional and $A$ is an open convex subset of $\mathscr {X}$, then $f(A)$ is an open interval.

Prove the following. If $\mathscr {X}$ is a real LCS and $A \subseteq \mathscr {X}$, then the closed linear span of $A$ is the intersection of all closed hyperplanes containing $A$.

Prove the following. Let $\mathscr {X}$ be a complex LCS and let $A$ and $B$ be two disjoint closed convex subsets of $\mathscr {X}$. If $B$ is compact, then there is a continuous linear functional $f : \mathscr {X} \to \mathbb {C}$, an $\alpha$ in $\mathbb {R}$, and an $\varepsilon > 0$ such that for $a$ in $A$ and $b$ in $B$,

Recall Theorem 3.7: if $\mathscr {X}$ is a real TVS and $A$ and $B$ are disjoint convex sets with $A$ open, then there is a continuous linear functional $f : \mathscr {X} \to \mathbb {R}$ and a real scalar $\alpha$ such that $f(a) < \alpha$ for all $a$ in $A$ and $f(b) \ge \alpha$ for all $b$ in $B$. If $B$ is also open, then $A$ and $B$ are strictly separated.

State and prove a version of this theorem for a complex TVS.

Recall Corollary 3.11: if $\mathscr {X}$ is a real LCS and $A \subseteq \mathscr {X}$, then $\overline{\operatorname {co}}(A)$ is the intersection of the closed half-spaces containing $A$.

State and prove a version of this corollary for a complex LCS.

Recall Corollary 3.12: if $\mathscr {X}$ is a real LCS and $A \subseteq \mathscr {X}$, then the closed linear span of $A$ is the intersection of all closed hyperplanes containing $A$.

State and prove a version of this corollary for a complex LCS.

Let $\mathscr {X}$ be a real LCS and let $A$ and $B$ be disjoint compact convex subsets of $\mathscr {X}$. Suppose $\mathscr {Y}$ is a subspace of $\mathscr {X}$ and $f_0 : \mathscr {Y} \to \mathbb {R}$ is a continuous linear functional such that $f_0(a) < 0$ for $a$ in $A \cap \mathscr {Y}$ and $f_0(b) > 0$ for $b$ in $B \cap \mathscr {Y}$. Show by an example that it is not always possible to extend $f_0$ to a continuous linear functional on $\mathscr {X}$ such that $f(a) < 0$ for $a$ in $A$ and $f(b) > 0$ for $b$ in $B$.

Let $P$ be the space of all polynomials in $x$ with real coefficients, and $U_+$ (resp., $U_-$) the subset of polynomials with positive (resp., negative) leading coefficient. Prove that the sets $U_+$ and $U_-$ are convex, but cannot be separated by a hyperplane.

Prove that disjoint convex closed sets $A$ and $B$ can be strictly separated by a hyperplane if one is compact. (This means that there exist a continuous linear functional $f$ and constants $c_1<c_2$ such that $f(x)\leq c_1$ on $A$ and $f(x)\geq c_2$ on $B$.)

Prove that every closed convex set in a real LCS $L$ is the intersection of some family of half-spaces of the form $f(x)\leq c$, where $f\in L'$, $c\in \mathbb {R}$.

Represent the unit ball of $L_p(n,\mathbb {R})$ as an intersection of countably many half-spaces.

Let $X$ be a locally convex topological vector space, $Y$ a closed subspace of $X$, and $x_{0}$ belong to $X \sim Y$. Show that there is a continuous functional $\psi : X \rightarrow \mathbb {R}$ such that

Show that the Hyperplane Separation Theorem may be amended as follows: the point $x_{0}$ may be replaced by a convex set $K_{0}$ that is disjoint from $K$ and the conclusion is that $K$ and $K_{0}$ can be separated by a closed hyperplane if $K_{0}$ is compact. If $K_{0}$ is open, there is a continuous linear functional $\psi$ on $X$ for which $\psi \left(x_{0}\right) < \inf_{x \in K} \psi (x)$ for all $x_{0} \in K_{0}$.

Prove the Hyperplane Separation Theorem for $H$, considered as a locally convex topological vector space with respect to the strong topology, by directly using Proposition 1.

Proposition 1: Let $K$ be a non-empty, closed, convex subset of a Hilbert space $H$ and $h_{0}$ be a vector in $H$ that lies outside of $K$. Then there is exactly one vector $h^{*} \in K$ that is closest to $h_{0}$, in the sense that $\left\| h_{0} - h^{*}\right\| = \operatorname {dist}_{K}(h_{0}) = \inf_{h \in K} \left\| h_{0} - h\right\|$.

Call a set $H \subset \mathbb {R}^{n}$ a hyperplane if there exist real numbers $a_{1}, \ldots , a_{n}, c$ (with $a_{i} \neq 0$ for at least one $i$) such that $H$ consists of all points $x = \left(x_{1}, \ldots , x_{n}\right)$ that satisfy $\sum a_{i} x_{i} = c$.

Suppose $E$ is a convex set in $\mathbb {R}^{n}$, with nonempty interior, and $y$ is a boundary point of $E$. Prove that there is a hyperplane $H$ such that $y \in H$ and $E$ lies entirely on one side of $H$. (State the conclusion more precisely). Suggestion: Suppose 0 is an interior point of $E$, let $M$ be the one-dimensional subspace that contains $y$, and apply Theorem 3.2.

Suppose $L^{2} = L^{2}([-1,1])$, with respect to Lebesgue measure. For each scalar $\alpha$, let $E_{\alpha }$ be the set of all continuous functions $f$ on $[-1,1]$ such that $f(0) = \alpha$. Show that each $E_{\alpha }$ is convex and that each is dense in $L^{2}$. Thus $E_{\alpha }$ and $E_{\beta }$ are disjoint convex sets (if $\alpha \neq \beta$) which cannot be separated by any continuous linear functional $\Lambda$ on $L^{2}$. Hint: What is $\Lambda \left(E_{\alpha }\right)$?

Suppose $X$ is a real vector space (without topology). Call a point $x_{0} \in A \subset X$ an internal point of $A$ if $A-x_{0}$ is an absorbing set.

Let $X$ be an infinite-dimensional Banach space and let $S = \left\{ x \in X: \left\| x\right\| = 1\right\}$ be the unit sphere of $X$. We want to cover $S$ with finitely many closed balls, none of which contains the origin of $X$. Can this be done in