If $X$ is separable, then the closed unit ball of $X^*$ is weak-star metrizable. Some calculus helps needed!

Question

Here is my effort to show this fact and I will use ball $X^*$ to denote the closed unit ball of $X^*$.

To show ball $X^*$ is weak star metrizable, we only have to show there is a metric $d$ on ball $X^*$ such that the topology induced by $d$ is the weak-star topology on ball $X^*$. Since $X$ is separable, ball $X$ is also separable. Thus there exists a countable dense subset $\{x_n\}$ in ball $X$. Now define the metric $d$ on ball $X^*$ by

$$d(x^*,y^*)=\sum_{n=1}^{\infty}\frac{|\langle x_n,x^*-y^*\rangle|}{2^n},$$

where $x^*,y^*\in\text{ball $X^*$}$. Let $T$ be the topology induced by $d$ and $wk^*$ be the weak-star topology on $\text{ball $X^*$}$. Then we need to show $T=wk^*$. And I try to use net convergence to show topology equivalence.

Let $x^*\in\text{ball $X^*$}$ and let $x_i^*$ be a net in $\text{ball $X^*$}$ such that $x_i^*\overset{wk^*}{\longrightarrow} x^*$. Then $\langle x_n,x_i^*\rangle\rightarrow\langle x_n,x^*\rangle$ for all $n$. Now Let $x^*\in\text{ball $X^*$}$ and let $x_i^*$ be a net in $\text{ball $X^*$}$ such that $x_i^*\rightarrow x^*$ in $(X,T)$. Then for each $\epsilon>0$, there exists $i_\epsilon\in\mathbb{N}$ such that $d(x_i^*,x^*)<\epsilon$ for all $i\geqslant i_\epsilon$; that is, for each $\epsilon>0$,

$$d(x_i^*,x^*)=\sum_{n=1}^{\infty}\frac{|\langle x_n,x_i^*-x^*\rangle|}{2^n}<\epsilon.$$

Here I want to show $x_i^*\overset{wk^*}{\longrightarrow} x^*$ if and only if $x_i^*\rightarrow x^*$ in $(X,T)$. But I forget some knowledge in Calculus. Can somebody help me to show this please? Thank you so much!!

Answer 1

Let $I$ denote the identity map from $(X^*,\text{weak*})$ onto $(X^*,d)$. Since $I$ is a bijection from a compact space (Banach-Alaoglu) into a Hausdorff space, we only need to show that $I$ is continuous.

To this end, fix a net $(x^*_\alpha)$ in $B_{X^*}$ such that $x^*_\alpha\to x^*$ in the weak* topology on $B_{X^*}$ for some $x^*\in B_{X^*}$. We want to show that $x^*_\alpha=I(x^*_\alpha)\to I(x^*)=x^*$ in the metric $d$; that is, we want: $$ \sum_{n=1}^\infty \frac{|\langle x_n,x_\alpha^*-x^*\rangle|}{2^n} \longrightarrow 0. $$ To this end, fix $\varepsilon>0$. Since each $x_n$ is in the unit ball of $X$ and $x^*$ and each $x^*_\alpha$ are in the unit ball of $X^*$, we deduce $$ \frac{|\langle x_n,x_\alpha^*-x^*\rangle|}{2^n} \le \frac{|x_\alpha^*(x_n)|+|x^*(x_n)|}{2^n} \le \frac{2}{2^n} = \frac{1}{2^{n-1}}. $$ Since $\sum_{n=1}^\infty2^{1-n}$ converges, there exists $N\in\mathbb{N}$ such that $\sum_{n=N+1}^\infty 2^{1-n}<\varepsilon/2$. Now, using the fact that $x_\alpha^*\to x^*$ in the weak* topology on $B_{X^*}$, for each $n\in\{1,\ldots,N\}$ there exists $\alpha_n$ in the directed set such that $$ |\langle x_n,x_\alpha^*-x^*\rangle| < \frac{2^n\varepsilon}{2N} $$ whenever $\alpha\ge\alpha_0$. Taking $\alpha_0$ in the direct set such that $\alpha_0\ge\alpha_1,\ldots,\alpha_N$, we obtain \begin{align*} \sum_{n=1}^\infty\frac{|\langle x_n,x_\alpha^*-x^*\rangle|}{2^n} &= \sum_{n=1}^N\frac{|\langle x_n,x_\alpha^*-x^*\rangle|}{2^n} +\sum_{n=N+1}^\infty\frac{|\langle x_n,x_\alpha^*-x^*\rangle|}{2^n} \\ &\le \sum_{n=1}^N \frac{\varepsilon}{2N} + \varepsilon/2 \\ &= \varepsilon \end{align*} whenever $\alpha\ge\alpha_0$. This completes the proof.

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A few comments on your post:

You write "Since $X$ is separable, ball $X$ is separable". This is a true statement, but only because $X$ is a metric space. In general topological spaces, subsets of separable spaces aren't necessarily separable.

It's important to show that $d$ is a metric. It is here that the density of $\{x_n : n\in\mathbb{N}\}$ in the unit ball of $X$ is actually needed.

You wrote "Then for each $\varepsilon>0$, there exists $i_\varepsilon\in\mathbb{N}$ such that $d(x_i^*,x^*)<\varepsilon$ for all $i\ge i_\varepsilon$." The problem is that $i_\varepsilon$ cannot be assumed to be a natural number, since your net may be indexed by some directed set other than $\mathbb{N}$.

Answer 2

You can do this directly: after checking that $d$ is a metric (it is), let $\tau$ be the topology induced by it. Then,

$(1). \ \sigma (X^*,X)\subseteq\tau :$

Let $S$ be the closed unit ball in $X^*,\ $ fix $f_0\in S$ and take $\left \{ f\in S:d(f_o,f) <\epsilon \right \}\subseteq \tau.$ There is an integer $k$ such that $\frac{1}{2^{k}}<\frac{\epsilon }{2}.$ Then, $\ U=\left \{ f\in S:|(f-f_0,x_j)|< \frac{\epsilon }{2};\ 0\le j\le k \right \}$ is weak* open in $S,$ contains $f_0$ and satisfies:

$\forall f\in U,\ d(f,f_0)=\sum_{n=0}^{k}\frac{(f-f_0,x_n)}{2^{n}}+\sum_{n> k}\frac{(f-f_0,x_n)}{2^{n}}<\frac{\epsilon }{2}+\frac{1}{2^{k}}<\epsilon.$

$(2).\ \tau \subseteq\sigma (X^*,X) :$

Take $U=\left \{ f\in S:|(f-f_0,y_j)|<\epsilon;\ 0\le j\le k \right \}$, a weak* neighborhood of $f_0.$ Let $M=\max \left \{ \|y_j\|:0\le j\le k \right \}.$ There are integers $n_j$ such that for any $\delta>0,\ \|x_{n_j}-y_j/M\|<\delta.$ And there is an $r>0,$ such that $M2^{n_j}r<\frac{\epsilon }{2}.$

Then, if $d(f,f_0)<r,$ we have of course, $\sum_{n=0}^{\infty }\frac{|(f-f_0,x_n)|}{2^{n}}<r$ so that $|(f-f_0,x_{n_j})|\le 2^{n_j}r$ and therefore,

$|(f-f_0,y_j)|=M|(f-f_0,y_j/M)|\le M|(f-f_0,x_{n_j})|+M|(f-f_0,x_{n_j}-y_j/M)|\le M(2^{n_j}r+2\delta)\le \epsilon /2+2M\delta.$

Thus, $f\in U$ as soon as $\delta$ is small enough.

Answer 3

Shorter, but less direct, proof. We show that dual ball $B_{X'}$ is homeomorphic to a subset of a metric space. Let $D$ be a countable dense subset in $X$. All continuous functions from $X$ to Hausdorf space (in particular to $ℝ$) are determined on $D$, hence $$ (B_{E'}, \sigma(E',E)) \ni f \mapsto f|_D \in ℝ^ℕ $$ is injective. This map is continuous ($ℝ^ℕ$ is equipped with the product topology, i.e. convergence = pointwise convergence). Indeed, let $f_\alpha \to f$ be a convergent net in $B_{E'}$ with weak* topology. This means that $$ f_\alpha(x) \to f(x) \quad (\forall x \in E). $$ In particular, for any $x \in D \subset E$ we have $$ (f_\alpha)|_D(x) = f_\alpha(x) \to f(x) = f|_D(x). $$ Hence $f\mapsto f|_D$ is a continuous injection. Moreover, $B_{E'}$ is weak* compact (Banach–Alaoglu theorem), hence this map is a homeomorphism onto its image which is a subset of a metrizable space $ℝ^ℕ$ (countable product of metric spaces is a metric).

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Comment. We swept under the rug the fact that countable product of metric spaces is a metric. This fact is usually shown by defining a metric analogous to the one described in question.