The central topic of the paper is the learnability of the recursively enumerable subspaces of V _∞/V , where V _∞ is the standard recursive vector space over the rationals with countably infinite dimension, and V is a given recursively enumerable subspace of V _∞. It is shown that certain types of vector spaces can be characterized in terms of learnability properties: V _∞/V is behaviourally correct learnable from text iff V is finitely dimensional, V _∞/V is behaviourally correct learnable from switching type of information iff V is finite-dimensional, 0-thin, or 1-thin. On the other hand, learnability from an informant does not correspond to similar algebraic properties of a given space. There are 0-thin spaces W ₁ and W ₂ such that W ₁ is not explanatorily learnable from informant and the infinite product (W ₁)^∞ is not behaviourally correct learnable, while W ₂ and the infinite product (W ₂)^∞ are both explanatorily learnable from informant.