I'm looking for a counterexample because I don't see any reason why this must be true.
F is a subspace of W
F is a subspace of V
$V \not\subset W$, $W \not\subset V$, and $V \neq W$
And there is no vector space that contains both V and W.
I was thinking about continuous functions and integrable functions... but they are all functions. No good. Maybe it's much too hard (too general) to show that there isn't a vector space that contains V and W as subspaces because there are just too many possibilities?
Things that don't work:
- $\forall a,b \in R$ vectors of the form (a,0,0) are a subspace of (a,b,0) and (a,0,b) but both of those are subspaces of R3.
- Is this the same subspace? integrable functions and continuous functions <- they are all functions