Let T be a linear operator on a finite-dimensional inner product space V.
1. If T is an orthogonal projection, prove that ||T(x)|| ||x|| for all x V. Give an example of a projection for which this inequality does not hold. What can be concluded about a projection for which the inequality is actually an equality for all x V?
2. Suppose that T is a projection such that ||T(x)|| ||x|| for x V. Prove that T is an orthogonal projection.