Orthogonal projection
= Orthogonal projection
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The orthogonal projection $P_F$ onto a closed subspace $F$ sends $x$ to the component in the decomposition $x=P_Fx+(I-P_F)x$ with $(I-P_F)x\in F^\perp$. It is a contraction: $\|P_Fx\|\leq\|x\|$.