The Separation of Convex Sets and Mazur's Theorem

For each natural number $n$, let $e_{n}$ denote the sequence in $\ell^{2}$ whose nth component is 1 and other components vanish. Define $E = \left\{ e_{n}+n \cdot e_{m} \mid n \text{ and } m \text{ any natural numbers}, m > n\right\}$. We showed that 0 is a point of closure of $E$ but no sequence in $E$ converges weakly to 0.

Consider the topological space $X = E \cup \left\{ 0\right\}$ with the weak topology. Find a function $f: X \rightarrow \mathbb {R}$ that fails to be continuous at 0 and yet has the property that whenever a sequence $\left\{ x_{n}\right\}$ in $E$ converges weakly to 0 , its image sequence $\left\{ f\left(x_{n}\right)\right\}$ converges to $f(0)$.

Find a subset of the plane $\mathbb {R}^{2}$ for which the origin is an internal point but not an interior point.

Let $X$ be a locally convex topological vector space and $V$ a convex, symmetric with respect to the origin (that is, $V = -V$) neighborhood of the origin. If $p_{V}$ is the gauge functional for $V$ and $\psi$ is a linear real-valued functional on $X$ such that $\psi \leq p_{V}$ on $X$, show that $\psi$ is continuous.

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

Let $X$ be a normed linear space and $W$ a proper subspace of $X^{*}$ that separates points. Let $\psi$ belong to $X^{*} \sim W$. Show that ker $\psi$ is strongly closed and convex but not $W$-weakly closed.

Let $X$ be a normed linear space. Show that the closed unit ball $B^{*}$ of $X^{*}$ is weak-* closed.

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 $x_{0} \in K_{0}$.

Show that the weak topology on an infinite dimensional normed linear space is not first countable.

Show that every weakly compact subset of a normed linear space is bounded with respect to the norm.

Let $Y$ be a closed subspace of a reflexive Banach space $X$. For $x_{0} \in X \sim Y$, show that there is a point in $Y$ that is closest to $x_{0}$.

Let $X$ be normed linear space, $W$ a finite dimensional subspace of $X^{*}$ and $\psi$ a functional in $X^{*} \sim W$. Show that there is a vector $x \in X$ such that $\psi (x) \neq 0$ while $\varphi (x) = 0$ for all $\varphi$ in $W$.

Let $X$ be a normed linear space. Show that any dense subset of $B^{*} = \left\{ \psi \in X^{*} \mid \left\| \psi \right\| \leq 1\right\}$ separates points in $X$.

Complete the final part of the proof of Proposition 24.

Proposition 24: Let $K$ be a convex subset of a linear space $X$ that contains the origin as an internal point, and let $p_{K}$ be the gauge functional for $K$, defined by $p_{K}(x) = \inf \left\{ \lambda > 0 \mid x \in \lambda \cdot K\right\}$. Then $p_{K}$ is subadditive and positively homogeneous. The proof given establishes only subadditivity and leaves the proof of positive homogeneity — that $p_{K}(\lambda x) = \lambda \cdot p_{K}(x)$ for all $\lambda > 0$ and $x \in X$ — as the part to complete.

Find an example of a bounded subset $A$ of a normed linear space $X, \mathcal{F}$ a set of functionals in $X^{*}$ containing $\mathcal{F}_{0}$ as a dense subset of $\mathcal{F}$ (dense in the sense of the norm topology on $X^{*}$) such that $\mathcal{F}$ and $\mathcal{F}_{0}$ generate different weak topologies for $X$, but the same weak topology for $A$.