Weakly sequentially continuous differentiable mappings

It is well known that a (linear) operator T ∈ L(X, Y ) between Banach spaces is completely continuous if and only if its adjoint T ∗ ∈ L(Y ∗ , X∗ ) takes bounded subsets of Y ∗ into uniformly completely continuous subsets, often called (L)-subsets, of X∗. We give similar results for differentiable mappings. More precisely, if U ⊆ X is an open convex subset, let f : U →Y be a differentiable mapping whose derivative f : U →L(X, Y ) is uniformly continuous on U-bounded subsets. We prove that f takes weak Cauchy U-bounded sequences into convergent sequences if and only if f takes Rosenthal U-bounded subsets of U into uniformly completely continuous subsets of L(X, Y ). As a consequence, we extend a result of P. Hájek and answer a question raised by R. Deville and E. Matheron. We derive differentiable characterizations of Banach spaces not containing 1 and of Banach spaces without the Schur property containing a copy of 1. Analogous results are given for differentiable mappings taking weakly convergent U-bounded sequences into convergent sequences. Finally, we show that if X has the hereditary Dunford–Pettis property, then every differentiable function f : U →R as above is locally weakly sequentially continuous.