H u K is a subspace of V iff H is contained in K or K is contained in H

Here, V is a vector space and H and K are subspaces.

If H is contained in K or K is contained in H, then H u K = K or H u K = H and in any case this is a subspace of V.

Now if H is not contained in K and K is not contained in H, then there is k in K\H and h in H\K.

Now both h, k lie in H u K. So let's look at h + k. If this were in H u K, it would be in H or in K. If it's in H, then k = (h + k) - h is in H, a contradiction. If it's in K, then h = (h + k) - k is in K, a contradiction. In any case we reach a contradiction so H u K is not closed under addition. In particular this means it cannot be a subspace.

A similar statement holds for ideals in a ring and subgroups of a group.