Linear algebra Orthogonal Projections

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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.
 

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