Infinite dimensional topological vectorspaces with dense finite dimensional subspaces

Question

Consider $\mathbb R$ as a $\mathbb Q$ vector space. Using the usual metric on $\mathbb R$, we find:

1. $\mathbb Q \subset \mathbb R$ is dense and one dimensional (indeed every non-zero subspace appears to be dense).
2. $\mathbb R$ is an infinite dimensional $\mathbb Q$ vectorspace.
3. Addition and scalar multiplication are continuous (addition on $\mathbb R \times \mathbb R$ and multiplication on $\mathbb Q \times \mathbb R$).
4. The metric is translation invariant.

Can there exist $\mathbb R$ (or $\mathbb C$) vector spaces that satisfy conditions 1-3? Meaning infinite dimensional topological vector spaces that have dense finite dimensional subspaces.

Is it possible to metricise these spaces? If so can condition 4 also be put in?

Answer 1

(1) Note that the topology on a finite-dimensional Hausdorff $\mathbf R$-vector space is uniquely determined and is always completely metriziable. As a complete metrizable space, is it closed in every topological vector space is it embedded into. Hence, finite-dimensional subspaces of Hausdorff topological vector spaces are always closed (and hence not dense if they are proper subspaces).

(2) If we drop the Hausdorff condition, an example can be given: Let $V$ any infinite dimensional $\mathbf R$ vector space and $\tau = \{\emptyset, V\}$ the trivial topology. Then every non-empty subset is dense (1 is true) and 2. and 3. are trivially true. Of course, a non-Hausdorff space cannot be metrizable, but it is pseudo-metrizable by the pseudo-metric $d(x,y)=0$ for all $x,y \in V$.