\(\ell^{p,\infty}\) has a complemented subspace isomorphic to \(\ell^ 2\) (Q1209556)

For a measure space \((\Omega,\Sigma,\mu)\) and \(1< p<\infty\), the weak \(L^ p\) space \(L(p,\infty,\mu)\) is the space of all measurable functions \(f\) such that \(\{\omega\mid| f(\omega)|> 0\}\) is \(\sigma\)-finite and \(\sup_ B\int_ B | f| d\mu/\mu(B)^{1-1/p}< \infty\), where the supremum is taken over all measurable sets \(B\) with \(0< \mu(B)< \infty\). \(\ell^{p,\infty}\) is the weak \(L^ p\) space in case \(\Omega= \mathbb{N}\) with the counting measure. The author proves the assertion claimed in the title.