# Weak$*$ topology is metrizable iff $X$ is finite dimensional.

Let $X$ be a normed $\mathbb K$-linear space . Then the weak$*$ topology is metrizable iff $X$ is finite dimensional.

We know that if X is finite dimensional then the weak topology on X is metrizable and also X is finite dimensional iff $X^*$ is finite dimensional.

From here can we approach for proof of the above theorem?

thank you..

I think you need $X$ to be a Banach space. Then first use the Banach-Steinhaus theorem to show that $(X^*,\sigma^*)$ is sequentially complete. If it is metrizable (hence a Frechet space) the open mapping theorem implies that the weak$^*$ topology coincides with the topology induced by the dual norm. For the unit ball $B$ of $X$ you thus get finitely many $x_1,\ldots,x_n\in X$ such that $\{x_1,\ldots,x_n\}^\circ \subset B^\circ$. Then apply the theorem of bipolars together with the fact that the absolutely convex hull $\Gamma(E)$ of a finite set is closed to get $B \subseteq \Gamma(\{x_1,\ldots,x_n\})$.

EDIT. As mentioned in the comment, for the space $X$ of scalar sequences with only finitely many non-zero terms endowed with the $\ell^2$-norm, the weak$^*$-topology on $X^*=\ell^2$ is the topology of coordinate-wise convergence which is metrizable.

• No.. I also asked my professor whether the space is Banach or not, he replied no. Commented Oct 24, 2017 at 15:58
• I do believe that for the countable dimensional space $X$ of real sequences with only finitely many non-zero terms endowed with the $\ell^2$- norm the dual (which is $\ell^2$) is metrizable in the weak$^*$ topology. Commented Oct 24, 2017 at 16:46
• Does your professor still say no? I would be interested in his or her argument. Commented Oct 25, 2017 at 12:56
• Any news from your professor? Commented Nov 2, 2017 at 8:54
• Sorry . I just forgot . He can give some hints but the question will remain same. Commented Nov 2, 2017 at 12:27

Hint: in infinite dimensions weakly open neighborhood of origin always contain a (maximal) vector subspace.

• This is the last part of my answer. What does your hint have to do with the metrizabiliy? Commented Oct 24, 2017 at 17:02
• Where is your answer? could you share your work here.. @Jochen Commented Oct 24, 2017 at 18:15