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**studentmath92** 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)|| $\displaystyle \leq$||x|| for all x$\displaystyle \in$ 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 $\displaystyle \in$ V?

2. Suppose that T is a projection such that ||T(x)|| $\displaystyle \leq $||x|| for x $\displaystyle \in$ V. Prove that T is an orthogonal projection.