If a vector space has a countable basis, can we construct it?

Question

Given that we cannot (?) exhibit a basis for a vector space of all real sequences (certainly that basis would be uncountable), but there is a countable basis for polynomials, i.e. $1,x,x^2,x^2,\ldots$, I would like to know,

If a vector space $V$ has a countable basis, can it be always constructed/enumerated (or whatever the correct term is)?

Answer 1

It is known that there is a computable vector space over $\mathbb{Q}$ with no computable basis. This also shows that we cannot prove constructively that every countable vector space over $\mathbb{Q}$ has a basis.