How do I prove that the given projection is orthogonal?

Asked Nov 24 '22 at 16:02

Active Nov 24 '22 at 16:02

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Prove that if $H$ is a Hilbert space and $P: H ⟶ H$ is a continuous linear projection of norm one, so $P$ is an orthogonal projection.

I know that the projection thing comes from the fact that $P^2=P$ and that the orthogonality thing comes from the fact that $x-Px⟂M$ where $M$ is a closed subspace of $H$.

asked Nov 24 '22 at 16:02

Andre

1

Does this answer your question? Show that if $|Q|=1$, then $Q$ is the orthogonal projection of $H$ onto $R(Q)$ (The range of $Q$) – Anne Bauval Nov 24 '22 at 16:08
@Anne Bauval, Do you know where the expression $(∥Q(y)∥^2−∥y∥^2)t^2+2t∥Q(y)∥^2$ comes from in the solution given in the link you mention? – Andre Nov 24 '22 at 17:01
From developping the inequality obtained just before: $(t+1)^2|Q(y)|^2\le|Q(y)|^2+t^2|y|^2.$ – Anne Bauval Nov 24 '22 at 17:10

How do I prove that the given projection is orthogonal?

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