1. The problem statement, all variables and given/known data Let P[tex]\in[/tex]L(V). If P^2=P, and llPvll<=llvll, prove that P is an orthogonal projection. 2. Relevant equations 3. The attempt at a solution I think that regarding llPvll<=llvll is redundant. For example, consider P^2=P and let v be a vector in V. Doesn't P^2=P kind of give it away by itself? I mean v=a1v1+...+amvm+...+anvn. Consider the subspace that P projects to whose dimension is less than V's. So for P^2 to =P, and P^2v=/=0, PPv=a1v1+...+amvm=Pv=a1v1+...+amvm. Notice how the scalars from 1 to m do not change to make P^2=P possible. Isn't it obvious enough from P^2=P that the length of the vector a1v1+..+amvm is < a1v1+...+amvm+...+anvn? I know that that's not were trying to prove, but why is llPvll<=llvll important when we know P^2=P?