Linear algebra Orthogonal Projections

[zcd]

Homework Statement

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

Homework Equations

$$T:V\to V$$ (V finite dimensional)
$$<Tx,y>=<x,T^* y>$$

general projection/idempotent operator:
$$V=R(T)\oplus N(T)$$
$$T^2=T$$


orthogonal projection:
$$R(T)=N(T)^{\perp}$$
$$T^2=T=T^*$$

The Attempt at a Solution

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

 

[lippyka]

I think it may be related to showing T=T^*. Any advice on how to approach this would be appreciated, thank you.