Let 𝒞(X,E) be the space of
continuous functions from the completely regular Hausdorff space X into the
Hausdorff locally convex space E, endowed with the compact-open topology. Our aim
is to characterize the 𝒞(X,E) spaces which have the following property: weak-star
and weak sequential convergences coincide in the equicontinuous subsets of 𝒞(X,E)′.
These spaces are here called Grothendieck spaces. It is shown that in the
equicontinuous subsets of E′ the σ(E′,E)- and β(E′,E)-sequential convergences
coincide, if 𝒞(X,E) is a Grothendieck space and X contains an infinite compact
subset. Conversely, if X is a G-space and E is a strict inductive limit of
Fréchet-Montel spaces 𝒞(X,E) is a Grothendieck space. Therefore, it is proved that
if E is a separable Fréchet space, then E is a Montel space if and only if there
is an infinite compact Hausdorff X such that 𝒞(X,E) is a Grothendieck
space.
