Compactness Regained: The Weak Topology

For $X$ a normed linear space with closed unit ball $B$, suppose that the function $f: B \rightarrow [-1,1]$ has the property that whenever $u, v, u+v$, and $\lambda u$ belong to $B, f(u+v) = f(u)+ f(v)$ and $f(\lambda u) = \lambda f(u)$. Show that $f$ is the restriction to $B$ of a linear functional on all of $X$ which belongs to the closed unit ball of $X^{*}$.

Let $X$ be a normed linear space and $K$ be a bounded convex weak-* closed subset of $X^{*}$. Show that $K$ possesses an extreme point.

Show that any non-empty weakly open set in an infinite dimensional normed linear space is unbounded with respect to the norm.

Use the Baire Category Theorem and the preceding problem to show that the weak topology on an infinite dimensional Banach space is not metrizable by a complete metric.

Is every Banach space isomorphic to the dual of a Banach space?

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

Show that the closed unit ball $B^{*}$ of the dual $X^{*}$ of a Banach space $X$ has an extreme point.

Let $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$ be two compact, Hausdorff topologies on a set $\mathcal{S}$ for which $\mathcal{T}_{1} \subseteq \mathcal{T}_{2}$. Show that $\mathcal{T}_{1} = \mathcal{T}_{2}$.

Let $X$ be a normed linear space containing the subspace $Y$. For $A \subseteq Y$, show that the weak topology on $A$ induced by $Y^{*}$ is the same as the topology $A$ inherits as a subspace of $X$ with its weak topology.

Argue as follows to show that a Banach space $X$ is reflexive if and only if its dual space $X^{*}$ is reflexive.

For $X$ a Banach space, by the preceding problem, if $X$ is reflexive, then so in $X^{*}$. Conclude that $X$ is not reflexive if there is a closed subspace of $X^{*}$ that is not reflexive. Let $K$ be an infinite, compact, Hausdorff space and $\left\{ x_{n}\right\}$ an enumeration of a countably infinite subset of $K$. Define the operator $T: l^{1} \rightarrow [C(K)]^{*}$ by

Show that $T$ is an isomorphism and therefore, since $l^{1}$ is not reflexive, neither is $T\left(l^{1}\right)$ and therefore neither is $C(K)$. Use a dimension counting argument to show that $C(K)$ is reflexive if $K$ is a finite set.

If $Y$ is a linear subspace of a Banach space $X$, we define the annihilator $Y^{\perp }$ to be the subspace of $X^{*}$ consisting of those $\psi \in X^{*}$ for which $\psi = 0$ on $Y$. If $Y$ is a subspace of $X^{*}$, we define $Y^{0}$ to be the subspace of vectors in $X$ for which $\psi (x) = 0$ for all $\psi \in Y$.

In a general topological space that is not metrizable a sequence may converge to more than one point. Show that this cannot occur for the $W$-weak topology on a normed linear space $X$, where $W$ is a subspace of $X^{*}$ that separates points in $X$.

Show that there is a bounded sequence in $L^{\infty }[0,1]$ that fails to have a weakly convergent subsequence. Show that the closed unit ball of $C[a, b]$ is not weakly compact.

Let $K$ be a compact metric space with infinitely many points. Show that there is a bounded sequence in $C(K)$ that fails to have a weakly convergent subsequence (see Problem 11), but every bounded sequence of continuous linear functionals on $C(K)$ has a subsequence that converges pointwise to a continuous linear functional on $C(K)$.

Show that the dual of an infinite dimensional normed linear space also is infinite dimensional.

Complete the last step of the proof of Theorem 10 by showing that the inequalities (11) imply that the metric $\rho$ induces the $W$-weak topology.

Theorem 10: Let $X$ be a normed linear space and $W$ a separable subspace of $X^{*}$ that separates points in $X$. Then the $W$-weak topology on the closed unit ball $B$ of $X$ is metrizable. The proof constructs a countable dense subset $\left\{ \psi_{k}\right\}_{k=1}^{\infty }$ of $B^{*} \cap W$ (where $B^{*}$ is the closed unit ball of $X^{*}$) and defines the metric

for which, for each natural number $n$,

Let $X$ be a Banach space, $W$ a closed subspace of its dual $X^{*}$, and $\psi_{0}$ belong to $X^{*} \sim W$. Show that if either $W$ or $X$ is reflexive, then there is a vector $x_{0}$ in $X$ for which $\psi_{0}\left(x_{0}\right) \neq 0$ but $\psi \left(x_{0}\right) = 0$ for all $\psi \in W$. Exhibit an example of an infinite dimensional closed subspace $W$ of $X^{*}$ for which this separation property fails.