Suppose you have a infinite dimensional vector space . Suppose . Can we have ?
E.g. can we partition an infinite dimensional vector space into finite dimensional ones?
I'm not sure how your two questions are related. The dimension theorem holds for infinite-dimensional vector spaces; but all that this tells us is that if then either the range or the kernel of must be infinite-dimensional.
It is impossible to partition a vector space into subspaces. If are subspaces and then we must have or . On top of that, the zero vector would have to be in every part.