Suppose $X$ is a topological vector space, $X^*$ is its topological dual space. Let the topology of $X^*$ is weak*-topology, Is $X^*$ complete?
Suppose $f_s$ is a Cauchy net in $X^*$, it is easy to see that $f=\lim f_s$ exists. We can prove that $f$ is linear, but I couldn't see if it is continuous.
No. For $X$ Hausdorff locally convex, the completion of $(X^*,\sigma^*)$ consists of all linear functionals on $X$ (the reason is that the semi-norms of $\sigma^*$ are determined by finite subsets of $X$ and on a finite dimensional subspace each linear functional is continuous and can be extended to a an element of $X^*$ by Hahn-Banach).
For example, if $X$ is an infinite dimensional Banach space the weak$^*$ dual is incomplete (this depends on the axiom of choice).
