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.

- 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.
- 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. - 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.)

equicontinuous weak$^*$ topology? Is it the finest topology on $X'$ which coincides with the weak$^*$ topology on all equicontinuous sets? A corollary of (or at least something closely related to) the Banach-Dieudonne theorem is stated in Köthe's book, page 273: Every precompact set in a metrizable locally convex space is contained in the absolutely convex hull of a sequence converging to $0$. This property also holds in strict LF-spaces $X=\lim X_n$ because every precompact set in $X$ is contained and precompact in some $X_n$. $\endgroup$ – Jochen Wengenroth May 4 '16 at 8:59