On the learnability of vector spaces

The central topic of the paper is the learnability of the recursively enumerable subspaces of V-infinity/V, where V-infinity is the standard recursive vector space over the rationals with countably infinite dimension, and V is a given recursively enumerable subspace of V-infinity. It is shown that certain types of vector spaces can be characterized in terms of learnability properties: V-infinity/V is behaviourally correct learnable from text iff V is finitely dimensional, V-infinity/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)(infinity) is not behaviourally correct learnable, while W-2 and the infinite product (W-2)(infinity) are both explanatorily learnable from informant.