A surjective summation operator with no Lipschitz right inverse

Abstract

We show that there exists a Banach space X X which contains closed subspaces Y Y and Z Z with Y + Z = X Y+Z=X such that the associated surjective summation operator Σ : Y × Z → X \Sigma \colon Y\times Z\to X defined by Σ ( y , z ) = y + z \Sigma (y,z)= y+z for y ∈ Y y\in Y and z ∈ Z z\in Z has no Lipschitz right inverse.