Completeness of the dual space of a Frechet Space I'm having trouble understanding completeness of duals of Fréchet spaces, I have been reading Meise and Vogt "Introduction to Functional Analysis" and something is not clear to me. So let $\mathcal{F}$ be a Fréchet space and $\mathcal{F}'$ the dual space, the space $\mathcal{F}'$ can be equipped with three different types of topology. They take the strong topology and show that the dual space is complete.
So my question is, is the space also complete if I take the weak star topology? If this is not the case what kind of conditions are needed for this to be true?
 A: Given a suitable family $\mathcal M$ of bounded subsets of $X$ you can endow the dual $X'$ with the locally convex topology defined by the seminorm $q_M(f)=\sup\{|f(x)|: x\in M\}$ for $M\in \mathcal M$ (the boudedness is used to have the supremum finite). If $(f_\alpha)_{\alpha\in I}$ is a Cauchy net with respect to this topology, you get convergence of all $(f_\alpha(x))_\alpha$ such that $x$ belongs to one of the sets $M$ (or even if $\lambda x\in M$ for some $M\in\mathcal M$ and $\lambda\neq 0$, because you can use the homogeneity of the functionals $f_\alpha$). If then $X$ is covered by (multiples of) the sets $M\in \mathcal M$, you get a pointwise limit $f(x)=\lim f_\alpha(x)$ which is a linear functional on $X$, and for all $M\in \mathcal M$ and $\varepsilon>0$ there is $\beta\in I$ such that, for all $\alpha \ge \beta$,
$$\sup\{|f_\alpha(x) - f(x)|:x\in M\}\le\varepsilon.$$
IF the pointwise limit $f$ is continuous on $X$, this would imply $f_\alpha\to f$ in $X'$ endowed with the topology described above. What you get from this condition is that $f$ is bounded on all $M\in \mathcal M$ and the question is thus: Does the boundedness of $f$ on all $M\in\mathcal M$ imply the continuity of the functional?
This depends of course on the family $\mathcal M$ and properties of the locally convex space $X$. If $X$ is metrizable and $\mathcal M$ is the class of all bounded sets, the answer is yes (as you read Meise and Vogt, this is the fact that metrizable spaces are bornological). However, if $\mathcal M$ is the family of all finite sets (you then get the weak$^*$ topology on $X'$), the assumption that $f$ is bounded on finite sets is trivially fulfilled for every function, and it is thus very unlikely that $(X',\sigma(X',X))$ is complete. 
This is the philosophy, to make a proof out of that you need a linear discontinuous functional on an arbitrary infinite-dimesnsioal Frechet space and you have to prove that there is a net of continuous functionals converging pointwise to it. The latter property (density of $X'$ in the algebraic dual $X^*$) follows from the fact that on finite-dimensional subspaces of $X$ all Hausdorff locally convex topologies coincide. 
