Question

Suppose U1,U2,..,Um are finite-dimensional subspace of V.

Prove that :

U1+U2+...+Um is finite dementional and

dim(U1+U2+...+Um)≤ dimU1 + dimU2 + ..... + dimUm

Answer 1

Suppose $u_1,u_2,...,u_m$ are finite-dimensional subspaces of V.

Thus, Each $u_j$ has a finite basis.

Concatenate these lists to get a spanning list of length $dim(u_1) + · · · + dim(u_m) \text{ for } u_1 + · · · + u_m.$

This shows that $u_1+· · ·+u_m$ is finite dimensional and since any spanning list can be reduced

to a basis then $dim(u_1 + · · · + u_m) ≤ dim(u_1) + · · · + dim(u_m).$