Is $\ell_1$ complemented in its double dual $\ell_1^{**}$? (i.e., in $\ell_\infty^*$?)

Question

Quick question, y'all. Is $\ell_1$ complemented in $\ell_1^{**}=\ell_\infty^*$?

Yes, I searched Google, and also the standard texts. I can't seem to find an answer, but surely this is known.

Thanks guys.

Answer 1

Daniel Fischer's comment says yes, since $\ell^1=c_0^*$. One can give an explicit projection from $(\ell^\infty)^*$ onto $\ell^1$ by saying $$P\Lambda=(\Lambda e_1,\Lambda e_2,,\dots).$$