Suppose $\ell_1=E\oplus F$ where $E$ and $F$ are linear subspaces with $E\cap F=\{0\}$, and $||x|| = ||x_1||+||x_2||$ for all $x\in \ell_1$ with $x=x_1+x_2$ with $x_1\in E$ and $x_2\in F$. Must $E$ and $F$ be norm closed?
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1$\begingroup$ You have an isometric isomorphism from $E\times F$ to $\ell_1$,where $E\times F$ is equipped with the sum norm. Since $E$ and $F$ are closed in $E\times F$, they are in $\ell_1$. $\endgroup$– user1688Commented Jan 26, 2017 at 9:19
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$\begingroup$ Having a decomposition such as this can be phrased as "$E$ is an L-summand in $\ell^1$". You have a corresponding continuous projection with image $E$ and kernel $F$. For further reading, I recommend this monograph: Harmand, P.; Werner, D.; Werner, W. $M$-ideals in Banach spaces and Banach algebras. Lecture Notes in Mathematics, 1547. Springer-Verlag, Berlin, 1993. viii+387 pp. ISBN: 3-540-56814-X MR1238713 $\endgroup$– anonymousCommented Jan 26, 2017 at 10:31
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$\begingroup$ Do you mean sequences with $\sum |a_i| < \infty $ by $\ell_1$? $\endgroup$– user83457Commented Jan 26, 2017 at 14:10
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I think that "yes", because the projections to these subspaces are continuous, and the range of a continuous projection in a Banach space is closed.
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$\begingroup$ It should be noted why the projections are continuous: $||P_1 x||=||x_1||\le||x_1||+||x_2||=||x||$, and same for $P_2$. $\endgroup$ Commented Jan 26, 2017 at 18:50