Separable Criterion: Weak-* Com./Seq. Comp. of Closed Unit Ball of X*

Separable Criterion for the Weak-* Compactness and Weak-* Sequential Compactness of the Closed Unit Ball of X*

Recall from the Helley's Theorem page that Helley's theorem states that if $X$ is a normed linear space and $X$ is separable then every bounded sequence in $X^*$ will weak-* converge.

We will now use Helley's theorem to show that if $X$ is a normed linear space and $X$ is separable then the closed unit ball of $X^*$ is weak-* compact and weak-* sequentially compact.

Theorem 1: Let $X$ be a normed linear space. If $X$ is separable then the closed unit ball of the topological dual $X^*$ is compact and sequentially compact in the weak-* topology.

In general the concepts of compactness and sequential compactness are different. If $X$ is a metric space, they are the same.

• Proof: Let $B_{X^*}$ denote the closed unit ball of $X^*$. That is:

(1)

\begin{align} \quad B_{X^*} = \{ f \in X^* : \| f \| \leq 1 \} \end{align}

• Since $X$ is separable, $B_{X^*} is separable. Let [[$ \{ x_n : n \in \mathbb{N} \}$ be a countable dense subset of $B_{X^*}$. Define a function $d : B_{X^*} \times B_{X^*} \to [0, \infty)$ for all $f, g \in B_{X^*}$ by:

(2)

\begin{align} \quad d(f, g) = \sum_{n=1}^{\infty} \frac{1}{2^n} | f(x_n) - g(x_n) | \end{align}

• We note that $d$ is well-defined. If $f, g \in B_{X^*}$ then $\| f \|, \| g \| \leq 1$ and so:

(3)

\begin{align} \quad \| f(x_n) - g(x_n) \| \leq \| f - g \| \| x_n \| \end{align}

• And so:

(4)

\begin{align} \quad d(f, g) = \| f - g \| \cdot \underbrace{\sum_{n=1}^{\infty} \frac{1}{2^n} \| x_n \|}_{\mathrm{converges \: since \:} \|x_n \| \leq 1} \end{align}

• We first show that $d$ is a metric on $X^*$.

• (1) Showing that $d(f, g) = 0$ if and only if $f = g$: Suppose that $d(f, g) = 0$. Then $f(x_n) = g(x_n)$ for all $n \in \mathbb{N}$. Let $x \in X$ and let $\epsilon > 0$ be given.. Since $\{ x_n : n \in \mathbb{N} \}$ is dense in $X$, there exists $n \in \mathbb{N}$ such that:

(5)

\begin{align} \quad | f(x) - f(x_n) | < \frac{\epsilon}{2} \quad \mathrm{and} \quad | g(x) - g(x_n) | < \frac{\epsilon}{2} \end{align}

• Therefore we have that:

(6)

\begin{align} \quad | f(x) -g(x) | \leq \underbrace{| f(x) - f(x_n) |}_{< \frac{\epsilon}{2}} + \underbrace{| f(x_n) - g(x_n) |}_{=0} + \underbrace{| g(x_n) - g(x) |}_{< \frac{\epsilon}{2}} < \epsilon \end{align}

• Since $\epsilon$ is arbitrary we see that $f(x) = g(x)$ for all $x \in X$. So $f = g$.

• On the other hand, if $f = g$ then trivially $d(f, g) = 0$.

• (2) Showing that $d(f, g) = d(g, f)$: This is clear since:

(7)

\begin{align} \quad d(f, g) = \sum_{n=1}^{\infty} \frac{1}{2^n} |f(x_n) - g(x_n)| = \sum_{n=1}^{\infty} \frac{1}{2^n}|g(x_n) - f(x_n)| = d(g, f) \end{align}

• (3) Showing that $d(f, g) \leq d(f, h) + d(h, g)$:

(8)

\begin{align} \quad d(f, g) &= \sum_{n=1}^{\infty} \frac{1}{2^n} |f(x_n) - g(x_n)| \\ &= \sum_{n=1}^{\infty} \frac{1}{2^n} |f(x_n) - h(x_n) + h(x_n) - g(x_n)| \\ & \leq \sum_{n=1}^{\infty} \frac{1}{2^n} | f(x_n) - h(x_n)| + \sum_{n=1}^{\infty} \frac{1}{2^n} |h(x_n) - g(x_n)| \\ & \leq d(f, h) + d(h, g) \end{align}

• So indeed, $d$ is a metric on $B_{X^*}$. Furthermore, this metric induces the weak-* topology on $B_{X^*}$.

• By Helley's Theorem, every sequence of continuous linear functionals on $B_{X^*}$ (which are bounded) has a weak-* convergent subsequence, and so $B_{X^*}$ is sequentially compact. Since $B_{X^*}$ is also a metric space, we have sequential compactness is equivalent to compactness. So $B_{X^*}$ is compact.$\blacksquare$

Important Note

The theorem above tells us that if $X$ is a normed linear space and $X$ is separable then the closed unit ball $B_{X^*}$ of $X^*$ is weak-* compact and weak-* sequentially compact.

Actually, if $X$ is just a normed linear space (and we don't assume $X$ is separable) then we can conclude that $B_{X^*}$ is weak-* compact. This is Alaoglu's Theorem. The same cannot be said for weak-* sequential compactness though. That is, if $X$ is not separable then $B_{X^*}$ may be weak-* sequentially compact OR may not be.