Define x,y∈P∩Q and also α∈F.
we have to prove that P∩Q=ϕ α(x+y)∈P∩Q
since P and Q are subspaces of V THEN , P =ϕ and Q=ϕ⟹P∩Q=ϕ
since P and Q are subspaces of V ⟹α(x+y)∈P and α(x+y)∈Q
Hence, α(x+y)∈P∩Q
Thus, we may conclude that, P∩Q is a subspace of V
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