Strictly singular quasi-linear maps (Q1599955)

A quasi-linear map \(F: Z\to Y\) is strictly singular if its restrictions to infinite-dimensional subspaces of \(Z\) are never trivial. Given a nontrivial quasi-linear map \(\ell_1\to\mathbb{R}\), then there exists a strictly singular map \(\ell_1\to \ell_\infty\). A ``transfer principle'' shows that for every infinite-dimensional subspace \(H\) of \(\ell_1\) the twisted sum \(\ell_1\oplus_{F_\theta} H\) is not locally convex. Conjecture: Strictly singular quasi-linear maps \(F: \ell_1(\Gamma)\to \mathbb{R}\) do not exist for sets \(\Gamma\) ``large'' enough.