Are open balls in the topological dual space $A^*$ weak-* open?

Here's an example of an argument. Non-empty weak* open sets always contain a translate of a finite-codimension sub-space. In particular for infinite dimensional spaces they may never be bounded.

Recall a sub-basis of the topology is given by sets of the form $\Psi^{-1}(U)$ with $U$ open in $\Bbb K$ and $\Psi$ a weak*-continuous functional. If $x$ is such that $x\in \Psi^{-1}(U)$ (ie the set is not empty) then clearly $x+\ker(\Psi)\subseteq \Psi^{-1}(U)$. Since $\Psi$ is a linear functional you have that $\ker(\Psi)$ has co-dimension one.

Now suppose you have a finite intersection $$\bigcap_{i=1}^n \Psi_i^{-1}(U_i)$$ if this is not empty there is some $x$ in it and then clearly $x+\bigcap_{i=1}^n \ker(\Psi_i)$ also is contained in this intersection. This intersection of kernels has co-dimension $n$, since its an intersection of $n$ co-dimension $1$ spaces.

This argument works for any topology induced by a space of linear functionals.