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?

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- Dec 9th 2009, 07:52 AMSamprasInfinite Dimensional Vector Spaces
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? - Dec 9th 2009, 08:02 AMBruno J.
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.