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## Homework Statement

Given T is a projection such that ||Tx||≤||x||, prove T is an orthogonal projection.

## Homework Equations

[tex]T:V\to V[/tex] (V finite dimensional)

[tex]<Tx,y>=<x,T^* y>[/tex]

general projection/idempotent operator:

[tex]V=R(T)\oplus N(T)[/tex]

[tex]T^2=T[/tex]

orthogonal projection:

[tex]R(T)=N(T)^{\perp}[/tex]

[tex]T^2=T=T^*[/tex]

## The Attempt at a Solution

I think the most straightforward approach would be to prove [tex]R(T)=N(T)^{\perp}[/tex], most likely by showing they contain each other. I'm having trouble seeing how the inequality of norms comes in though.