Lifting problems for normed spaces

Abstract

A classical theorem of G. Köthe states that the Banach spaces $X$ with the property that all bounded linear maps $X\to Y$ into an arbitrary Banach space $Y$ can be lifted with respect to bounded linear surjections onto $Y$ are up to topological linear isomorphism precisely the spaces ${\ell }^1\left(A\right)$. We extend this result to the category of normed linear spaces and bounded linear maps. This answers a question raised by A. Ya. Helemskiĭ.