If E is a projection on V with range W and suppose $\displaystyle \|Ex\| \le \|x\|$ for all $\displaystyle x \in V$, prove that E is in fact an orthogonal projection on W.
If E is not orthogonal on W, then there exist some other subspace, U, such that E is orthogonal on U. Let x be a vector in U and show that ||Ex||> ||x||.