Convex hulls and closed convex hulls in topological vector spaces

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.

If $\mathcal{X}$ is a LCS and $K_1, \dots , K_n$ are compact convex subsets of $\mathcal{X}$, then $\overline{\co }(K_1 \cup \dots \cup K_n) = \co (K_1 \cup \dots \cup K_n)$ and this convex hull is compact.

If $\mathscr {X}$ is a Banach space and $K$ is a compact subset of $\mathscr {X}$, prove that $\overline{\co }(K)$ is compact. (This will be proved in Theorem VI.4.8 below.)

In the proof of (13.4) (where $K$ is a weakly compact subset of a Banach space $\mathscr {X}$, $M(K) = C(K)^*$, and $T : M(K) \to \mathscr {X}$ is defined by $T(\mu ) = x_\mu$, the element of $\mathscr {X}$ such that $\left\langle x_\mu , x^* \right\rangle = \int_K \left\langle x, x^* \right\rangle \, d\mu (x)$ for all $x^*$ in $\mathscr {X}^*$), if $\mathscr {P} =$ the probability measures on $K$, show that $T(\mathscr {P}) = \overline{\co }(K)$.

Let $A$ be any set in a linear space $L$, $c_1(A)$ the intersection of all the convex sets in $L$ containing $A$, and $c_2(A)$ the collection of all vectors of the form $x=\sum_{i=1}^n\tau_i x_i$, where $x_i\in A$ and the coefficients $\tau_i$ are nonnegative and have the property that $\sum_{i=1}^n\tau_i=1$. Prove that $c_1(A)=c_2(A)$. This set is called the convex hull of $A$ and denoted by $c(A)$.

Prove that in every LCS the convex hull of a bounded set is bounded. Give an example showing that this property may fail to hold in an arbitrary LTS.

The convex hull of a set $A$ in a vector space $X$ is the set of all convex combinations of members of $A$, that is, the set of all sums

in which $x_{i} \in A, t_{i} \geq 0, \sum t_{i} = 1 ; n$ is arbitrary. Prove that the convex hull of $A$ is convex and that it is the intersection of all convex sets that contain $A$.

Let $X$ be a topological vector space. All sets mentioned below are understood to be the subsets of $X$. Prove the following statements.

Define functionals $\Lambda_{m}$ on $\ell^{2}$ (see Exercise 5) by

Define $x_{n} \in \ell^{2}$ by $x_{n}(n) = 1 / n, x_{n}(i) = 0$ if $i \neq n$. Let $K \subset \ell^{2}$ consist of $0, x_{1}, x_{2}$, $x_{3}, \ldots$ Prove that $K$ is compact. Compute $\Lambda_{m} x_{n}$. Show that $\left\{ \Lambda_{m} x\right\}$ is bounded for each $x \in K$ but $\left\{ \Lambda_{m} x_{m}\right\}$ is not. Convexity can therefore not be omitted from the hypotheses of Theorem 2.9.

Choose $c_{n} > 0$ so that $\sum c_{n} = 1, \sum n c_{n} = \infty$. Take $x = \sum c_{n} x_{n}$. Show that $x$ lies in the closed convex hull of $K$ (by definition, this is the closure of the convex hull) and that $\left\{ \Lambda_{m} x\right\}$ is not bounded.

Show that the convex hull of $K$ is not closed.

Let $\left\{ u_{1}, u_{2}, u_{3}, \ldots \right\}$ be a sequence of pairwise orthogonal unit vectors in a Hilbert space. Let $K$ consist of the vectors 0 and $n^{-1} u_{n}(n \geq 1)$. Show that

If $0 < p < 1$, show that $\ell^{p}$ contains a compact set $K$ whose convex hull is unbounded. This happens in spite of the fact that $\left(\ell^{p}\right)^{*}$ separates points on $\ell^{p}$; see Exercise 5. Suggestion: Define $x_{n} \in \ell^{p}$ by

Let $K$ consist of $0, x_{1}, x_{2}, x_{3}, \ldots$ If

show that $\left\{ y_{N}\right\}$ is unbounded in $\ell^{p}$.

Suppose that $A_{1}, \ldots , A_{n}$ are convex sets in a vector space $X$. Prove that every $x \in \operatorname {co}\left(A_{1} \cup \cdots \cup A_{n}\right)$ can be represented in the form

with $a_{i} \in A_{i}$ and $t_{i} \geq 0$ for all $i, \sum t_{i} = 1$.

Let $C(I)$ be the Banach space of all continuous complex functions on the closed unit interval $I$, with the supremum norm. Let $M = C(I)^{*}$, the space of all complex Borel measures on $I$. Give $M$ the weak*-topology induced by $C(I)$.

For each $t \in I$, let $e_{t} \in M$ be the "evaluation functional" defined by $e_{t} f = f(t)$, and define $\Lambda \in M$ by $\Lambda f = \int_{0}^{1} f(s) d s$.

Suppose $X$ is a vector space, $E \subset X, T: \operatorname {co}(E) \rightarrow X$ is affine, and $T(E) \subset E$. Prove that $T(\operatorname {co}(E)) \subset \operatorname {co}(E)$. (This was tacitly used in the proof of Theorem 5.11).

Prove the statement about convex hulls made in the last sentence of the proof of Theorem 8.4.