Extensions of smooth mappings into biduals and weak continuity

We study properties of uniformly differentiable mappings between real Banach spaces. Among our main results are generalizations of a number of classical results for linear operators on L∞L∞-spaces into the setting of uniformly differentiable mappings. Denote by BXBX the closed unit ball of a Banach space XX. Let XX be a L∞,λL∞,λ-space, λ≥1λ≥1, and let YY be a Banach space. Let T:BX→YT:BX→Y be a continuous mapping which is uniformly differentiable in the open unit ball of XX. Assuming that TT is weakly compact, then TT can be extended, preserving its best smoothness properties, into the mapping from the 1λ-multiple of the unit ball of any superspace of the domain space XX into the same range space YY. We also show that TT maps weakly Cauchy sequences from λBXλBX into norm convergent sequences in YY. This is a uniformly smooth version of the Dunford–Pettis property for the L∞,λL∞,λ-spaces. We also show that a uniformly differentiable mapping TT, which is not necessarily weakly compact, still maps weakly Cauchy sequences from λBXλBX into norm convergent sequences in YY, provided Y∗∗Y∗∗ does not contain an isomorphic copy of c0c0.We prove that for certain pairs of Banach spaces the completion of the space of polynomials equipped with the topology of uniform convergence on the bounded sets (of the functions and their derivatives up to order kk) coincides with the space of uniformly differentiable (up to order kk) mappings.Our work is based on a number of tools that are of independent interest. We prove, for every pair of Banach spaces X,YX,Y, that any continuous mapping T:BX→YT:BX→Y, which is uniformly differentiable of order up to kk in the interior of BXBX, can be extended, preserving its best smoothness, into a bidual mapping T̃:BX∗∗→Y∗∗. Another main tool is a result of Zippin’s type. We show that weakly Cauchy sequences in X=C(K)X=C(K) can be uniformly well approximated by weakly Cauchy sequences from a certain C[0,α]C[0,α], αα is a countable ordinal, subspace of X∗∗X∗∗.