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Here I attach the theorem and comment that I am confused about: enter image description here

enter image description here

I am confused about the comment right below the definition. From what I learned in topology, all projection maps are continuous. If this is true, why do we need Theorem 2.10 to obtain this result?

Mathematics_Beginner
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    The projection mapping $P:X\to X$, by definition satisfies $P\circ P=P.$ The continuity does not follows automatically. For example, concerning Theorem 2.10, if $G+L$ is not closed, then the conclusion does not hold and the projection operators described in Definition are not continuous. – Ryszard Szwarc Feb 21 '22 at 07:21

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One can show that on every Banach space $X$, there exists a discontinuous linear functional $f:X\to\mathbb{C}$. Let $x\in X$ be such that $f(x)=1$. Define $P:X\to X$ by $Py = f(y)x$. It is not difficult to show that $P$ is linear and $P^2=P$. Since $f$ is discontinuous, so is $P$.

Onur Oktay
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