When are metrizable linear subspaces of topological vector spaces finite-dimensional?

We study conditions that imply finite-dimensionality of metrizable or Frechet-Urysohn linear subspaces of topological vector spaces (TVS). We consider the classes ${\mathcal{D}}_M$ and ${\mathcal{D}}_{FU}$ of TVS: a space $L$ belongs to ${\mathcal{D}}_M$ (respectively, ${\mathcal{D}}_{FU}$) if every metrizable (respectively, Frechet-Urysohn) linear subspace of $L$ is finite-dimensional. Although the classes of metrizable spaces and of Frechet-Urysohn spaces differ substantially, the classes ${\mathcal{D}}_M$ and ${\mathcal{D}}_{FU}$ are much closer to each other: in the Hrusak-Ramos-Garcia model, in which Malykhin’s problem is solved, ${\mathcal{D}}_M={\mathcal{D}}_{FU}$, whereas under MA+not CH, as shown in the paper, ${\mathcal{D}}_M\mathrm{\ne }{\mathcal{D}}_{FU}$. For a locally convex space (LCS) $L$ we denote by $L_w$ the space $L$ with its weak topology and by $L_{w^{\ast }}^{\mathrm{\prime }}$ the dual space with the weak${}^{\ast }$ topology. This paper investigates which LCS $L$ have the property that $L_w$ and $L_{w^{\ast }}^{\mathrm{\prime }}$ belong to ${\mathcal{D}}_M$ or ${\mathcal{D}}_{FU}$. Although the classes are close, different methods are used to study them: topological-algebraic ones for ${\mathcal{D}}_M$ (a criterion is obtained) and methods of measure theory and probability for ${\mathcal{D}}_{FU}$ (a sufficient condition is obtained). We prove that if an LCS $L$ admits a continuous norm, then $L_w\in {\mathcal{D}}_{FU}$; in particular, if $L$ admits a continuous norm and $L_w$ is Frechet-Urysohn, then $L$ is finite-dimensional. If the topology of an LCS $L$ is coarser than the topology induced by some complete norm on $L$ (for example, if $L$ is Banach), then $L_{w^{\ast }}^{\mathrm{\prime }}\in {\mathcal{D}}_{FU}$; completeness of the norm is essential here. The methods also apply to spaces of real-analytic functions and to the space of polynomials with the topology of pointwise convergence: these also belong to ${\mathcal{D}}_{FU}$.