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I have a quick question, If $E$ is a Banach space and $H$ is a closed subspace of $E$, could we affirm this proposition:

If exists a linear continuous function $S:E\to H$ such that $S\circ i =Id_{_H}$ (with $i:H\hookrightarrow E$), then $H$ is complemented in $E$.

I don't need the proof, simply I need to know if this is true or false, thanks.

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1 Answer 1

In this case $H\oplus \ker(S)=E$. The proof is as follows: if $h\in H\cap \ker(S)$ then $0=S(h)=h$, the second equality follows from $S\circ i=Id_H$. Given $e\in E$, then $e=(e-S(e))+S(e)$ where $e-S(e)\in \ker(S)$ and $S(e)\in H$.

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