Definition of countable dimension of vector space

Question

It's a well known that Hausdorff locally convex space $(X, P)$, where $P$ is a family of seminorms which generate topology on $X$ is metrizable iff $P$ is equivalent to an at most countable subfamily $P_0 \subset P$. Using this i am trying to prove that the weak topology on $X$ generated by $Y$ where $(X,Y)$ is a dual pair ($=$dual system) is metrizable iff the dimension of $Y$ is at most countable, so here comes my problem, in which sense we understand the dimension of Y is at most countable? Is it about Hamel basis, or about Schauder basis, or in another way, because the are several ways to define a dimension of vector space?

Answer 1

The metrisability is due to the countable local base such a space needs to have.

Countable dimension is just a linear property: a countable set of vectors that (finitely) generates the space, so a countable Hamel basis. There are many metrisable locally convex spaces that are not countable dimensional, like $C([0,1])$ and $\Bbb R^\omega$, so it's not coupled to dimension. In fact a completely metrisable such space cannot be countable dimensional, because Baire fails almost always in such spaces.