The Banach-Dieudonné theorem states that if $X$ is a metrizable locally convex Hausdorff space then the equicontinuous weak-\* topology $ew^\*$ on $X'$ coincides with the topology of precompact convergence and is therefore a locally convex topology. ($ew^\*$ is the final topology on $X'$ coinduced by the inclusions of the equicontinuous sets when equipped with the weak-\* topology $w^\*$. Note that $ew^\*$ is a priori not the locally convex final topology of these inclusions!) If $X$ is complete and thus Fréchet then it also coincides with the topology of compact convergence.


Is this also true in the case that $X$ is a strict inductive limit of a sequence of Banach or Fréchet spaces? In other words, is the equicontinuous weak-\* topology on the dual of an LB- or LF-space locally convex or at least linear?


**EDIT**: I think to have found a counterexample which I try to sketch.
Consider the LF-space $\mathcal{D} := C^\infty\_c(\mathbb{R})$ of test functions with the locally convex inductive limit topology and its dual $\mathcal{D}'$ - the space of distributions.


1. Since $\mathcal{D}$ is Montel, the strong dual $\mathcal{D}'\_\beta$ is Montel and thus a sequence in $\mathcal{D}'$ is strongly convergent iff it is weakly\*-convergent.
2. It is not hard to see that if $X$ is the strict inductive limit of *separable* Fréchet spaces then $ew^\*$ is a sequential topology and has the same convergent sequences as $w^\*$. Thus, $ew^\*$ is the sequential coreflection of $w^\*$. (The separability of the Fréchet spaces induce separability of $X$ which in turn is used for the (equicontinuous) polars of a neighborhood base of $0$ in $X$ to be metrizable and thus sequential. I can sketch a more detailed proof.) The space $X = \mathcal{D}$ satisfies these assumptions.
3. Dudley, "Convergence of Sequences of Distributions" (1971) has shown that $\mathcal{D}'\_\beta$ is not sequential and that the topology of all sequentially (strongly) open sets (which by 1. and 2. coincides with $ew^\*$) is not a vector topology (addition is merely jointly sequentially continuous).


For my applications it is rather of interest, whether for the LB-space $X = C\_c(\mathbb{R})$ the $ew^\*$-topology on its dual $X'$ (the space of real Radon measures) is a vector topology. We can't use the above proof since point 1. is not satisfied for $X$, i.e. a weakly\* convergent sequence in $X'$ needs not be strongly convergent. (Dudley has stated in his paper that $X'\_\beta$ is not sequential, but I can't use this fact to check the linearity of the $ew^\*$-topology.)



Is the equicontinuous weak-star topology locally convex on the dual of an LF-space?
An interesting paper on spaces for which the $ew^\*$ topology coincides with the topology of precompact convergence is [S-spaces and the open mapping theorem](http://msp.org/pjm/1962/12-1/p23.xhtml) by Taqdir Husain. He calls such spaces S-spaces, a term that doesn't appear to have become standard.


Proposition 2 of that paper shows that a complete locally convex space is an S-space if and only if the $ew^\*$ topology is locally convex. Corollary 2.1 that precedes it states that a complete S-space is always B-complete (i.e. fully complete in the sense of Ptak). Section 9 is on inductive limits, and notes that Grothendieck had already given an example of an LF space that is not B-complete. Unfortunately the sufficient condition he gives for strict inductive limits to be S-spaces can never apply to limits of infinite-dimensional Banach spaces.


