Are projections always continuous? If they are, is there simple way to prove it? If P:V->V is a projection, I can see that P(V) is a subspace, and restriction of P to this subspace is the identity, and it seems intuitively clear that vectors outside this subspace are always mapped to shorter ones, but I don't know how to prove it. If V was a Hilbert space, and we knew P(V) is closed, then I could prove this using the projection theorem. However only way to prove that P(V) is closed, that I know, is to use continuity of P.