Let $X$ be a Banach space. $P∈B(X)$ satisfies $P^2=P$. How to prove that $X=\ker P⊕P(X)$ ? Thanks for your help!
2026-03-29 05:43:38.1774763018
Let $X$ be a Banach space. $P∈B(X)$ satisfies $P^2=P$. Prove that $X=\ker P⊕P(X)$
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Hint
$$x-P(x)\in \ker(P)$$ for all $x$.