The Banach-Dieudonné theorem states that if $X$ is a metrizable locally convex Hausdorff space then the equicontinuous weak-* topology on $X'$ coincides with the topology of precompact convergence and is therefore a locally convex topology. 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?