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 ℓ 1 and of Banach spaces without the Schur property containing a copy of ℓ 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.