Metrizable and Fréchet--Urysohn subgroups and linear subspaces of topological vector spaces

For a normed space $E$, we denote by $E_w$ and $E_{w^{\ast }}^{\mathrm{\prime }}$ the space $E$ and its dual $E^{\mathrm{\prime }}$ endowed with the weak topology $w$ and the weak${}^{\ast }$ topology $w^{\ast }$, respectively. It is well known that for a Banach space $E$, both spaces $E_w$ and $E_{w^{\ast }}^{\mathrm{\prime }}$ are Fréchet–Urysohn topological spaces if and only if $E$ is finite-dimensional. This statement remains true for all linear subspaces of $E_w$ and $E_{w^{\ast }}^{\mathrm{\prime }}$. Moreover, for any normed space $E$, every Fréchet–Urysohn subgroup of $E_w$ is locally finite-dimensional; that is, it has an open subgroup contained in a finite-dimensional linear subspace. Similarly, for any Banach space $E$, every Fréchet–Urysohn subgroup of $E_{w^{\ast }}^{\mathrm{\prime }}$ is locally finite-dimensional. Analogous results are obtained for the free locally convex space $L(X)$, its weak version $L_p(X)$, and the free topological vector space $V(X)$ over a Tychonoff space~$X$. The same conclusions hold for Baire subgroups and Baire linear subspaces of $E_w$, $E_{w^{\ast }}^{\mathrm{\prime }}$, $L(X)$, $L_p(X)$, and $V(X)$. We emphasize that even for metrizable subgroups, these results are new and nontrivial. In fact, all results are established for the much wider family of $\kappa$-Fréchet–Urysohn subgroups of topological vector spaces. The class of $\kappa$-Fréchet–Urysohn spaces is the natural framework in which the proofs of our results work.