Sergeib's question asks about vector spaces without a natural basis.

Actually, I would claim (apparently in accord with many comments and answers to Sergeib's question ) that this is the default situation and that it is a rare event if a naturally occurring vector space *does* have a natural basis or even if it has any basis that can be described explicitly: the notorious $\mathbb Q \; -$ vector space $\mathbb R$ being the foremost example of this impossibility of exhibiting an explicit basis . Of course tautological examples like polynomial rings don't contradict this thesis, since they are defined as free vector spaces on some set!

In fact, the only example I can see without thinking twice of a natural vector space with a big natural non-tautological basis is the rational function field $k(X)$ seen as a $k$ - vector space .

Namely, consider a field $k$ and the set of monic polynomials $f_i(X) \; (i \in I)$ irreducible over $k$. Then $k(X)$ has the following $k$ - basis:

the mononomials $X^k (k\in \mathbb N) $ and the rational fractions $\frac{X^m}{f_i(X)^s}$ (with $i\in I,\; s>0$ and $ m< deg f_i $ )

In particular this natural basis is non denumerable if $k$ is non denumerable..

**Question** Which other vector spaces do you know for which some (preferably big) explicit basis can be given, but which are not clearly constructed as free vector spaces over a set?