What is Banach-Dieudonne theorem?

Question

Let $X$ be a separable Banach space and $X^*$ be its dual and $w^*$ be weak$^*$ topology on $X^*$. Let $B(X^*)$ denotes closed unit ball of $X^*.$

I'm reading a research paper in which we want to prove that a subset $K$ of $X^*$ is closed in $w^*$-topology. According to the paper by Banach-Dieudonne theorem, it is enough to prove that $B(X^*)\cap K$ is closed in $w^*$-topology. I want to know about the Banach-Dieudonne theorem.

Answer 1

The Banach-Dieudonné theorem states:

Let $X$ be a Banach space and $A$ be a convex set in the dual $X^{*}$. If $A\cap nB_{X^{*}}$ is w$^{*}$-closed for every $n \in {\mathbb N}$ then $A$ is w$^{*}$-closed.

Here, $B_{X^{*}}$ is the unit ball of the dual space $X^{*}$ as usual, which you're notating as $B(X^{*})$.

Translating this into English, this says that a convex set of continuous linear functionals ($A$) that has a w$^{*}$-closed intersection with every integer scaling of the unit ball is w$^{*}$-closed, and from that it's easy to see that the same holds true for any closed ball in $X^{*}$ (i.e. the unit ball can be translated to any point $x^* \in X^*$). The proof of the theorem is not especially hard, but is about a page or two in length. It's a nice exercise in using polars.

The convexity of the set $A$ in the theorem is essential. Take, for example, $\ell_2$ as our Banach space with the usual basis $\{e_i\}_{i \in {\mathbb N}}$ and let $x_i :=\sqrt{i} \cdot e_i$. Set $A:=\{x_i \mid i \in {\mathbb N}\}$. Since $0$ does belong to the w$^*$-closure of $A$ but $A\cap nB_{\ell_2}$ is finite (and so w$^*$-closed) for each $n\in \mathbb N$ the theorem does not hold.

The particular utility of the Banach-Dieudonné theorem comes from the following corollary (see M. Fabian et al, Banach Space Theory, pp.$124-125$ for a proof, for example):

Let $X$ be a Banach space and $F \in X^{**}$. The following are equivalent:

1. $F$ is w$^*$-continuous
2. For every $f \in X^*$ there is an $x\in X$ with $F(f) = f(x)$
3. The restriction of $F$ to $B_{X^*}$ is w$^*$-continuous
4. $F^{-1}(0)\cap B_{X^*}$ is w$^*$-closed

Note that $X$ is not required to be reflexive, but by $2$ of the corollary we have that $F$ must belong to $X$, so we can obtain some of the benefits of reflexivity for this $F$.