Reflexivity and Weak Compactness: Kakutani's Theorem

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 $\psi \in Y$.