# Math Help - Infinite Dimensional Vector Spaces

1. ## Infinite Dimensional Vector Spaces

Suppose you have a infinite dimensional vector space $V$. Suppose $T \in \mathcal{L}(V,W)$. Can we have $\dim V = \dim \text{null} \ T + \dim \text{range} \ T$?

E.g. can we partition an infinite dimensional vector space into finite dimensional ones?

2. 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 $\dim V = \infty$ then either the range or the kernel of $T$ must be infinite-dimensional.

It is impossible to partition a vector space into subspaces. If $W_1, W_2$ are subspaces and $W_1 \cup W_2 = V$ then we must have $W_1 \subset V$ or $W_2 \subset V$. On top of that, the zero vector would have to be in every part.