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?



$L$-decompositions of $\ell_1$
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.


