Given two subspaces H and K of a vector space V , the sum of H and K, written as H + K, is the set of all vectors in V that can be written as a sum of a vector in H and a vector in K, that is,

H + K = {w | w = u + v for some u H and some v K}

Show that H + K is a subspace of V (Hint: do not forget to show that H + K is indeed a subset of V ).