Extensions of smooth mappings into biduals and weak continuity
Results on extensions of multilinear mappings and polynomials between Banach spaces to their biduals have appeared numerous times in the literature, probably starting with the results of \textit{R. Arens} [Proc. Am. Math. Soc. 2, 839--848 (1951; Zbl 0044.32601)], where bilinear mappings are extended. The classical Aron-Berner extension results [\textit{R. M. Aron} and \textit{P. D. Berner}, Bull. Soc. Math. Fr. 106, 3--24 (1978; Zbl 0378.46043)] build extension operators for the space of multilinear forms and then use Taylor expansions to extend holomorphic functions of bounded type from a Banach space to its bidual. The present paper presents a number of extension results in the context of uniformly differentiable mappings between real Banach spaces -- although many of the results can be carried on, as the authors mention, to the complex Banach space case or, even, to the locally convex setting. They cover the classical cases mentioned above and many other previous results in the literature. For example, the following generalization of a result of Lindenstrauss is presented: Let \(X\) be an \({\mathcal L}_{\infty,\lambda}\)-space for \(\lambda\geq 1\), and let \(Y\) be a Banach space. Let \(T:B_{X}\rightarrow Y\) be a weakly compact, uniformly smooth (in the interior of \(B_X\)) operator. Then \(T\) can be extended, preserving its best smoothness properties, into a mapping from the \((1/\lambda)\)-multiple of the unit ball of any superspace. Moreover, \(T\) maps weakly Cauchy sequence into norm-convergent ones, giving a smooth version of the Dunford-Pettis property for the \({\mathcal L}_{\infty}\)-spaces that is a generalization of the work of Dunford, Pettis, Grothendieck, Lindenstrauss and Pełczyński, and Ryan, who proved the result for polynomials. It is also shown that a uniformly differentiable mapping \(T\), not necessarily weakly compact, still maps weakly Cauchy sequences from \(\lambda B_X\) into norm-convergent sequences in \(Y\), provided that \(Y^{**}\) does not contain an isomorphic copy of \(c_0\). Adding the assumption that \(X\) does not contain an isomorphic copy of \(\ell_1\) then \(T\) is always weakly compact, provided that \(Y^{**}\) does not contain an isomorphic copy of \(c_0\), a generalization of Pełczyński's result for polynomials. Other applications involve a generalization of the Stone-Weierstrass theorem: For certain pairs of Banach spaces, the completion of the space of polynomials equipped with the topology of uniform convergence on bounded sets (of the functions and their derivatives up to order \(k\)) coincides with the space of uniformly differentiable (up to order \(k\)) mappings, a generalization of the classical de la Vallée Poussin theorem, and the work of Aron and Prolla. A number of tools -- of independent interest -- are developed. For example, it is proven that any uniformly differentiable (of order up to \(k\)) mapping \(T:B_{X}\rightarrow Y\) can be extended -- preserving its best smoothness -- into a bidual mapping \(\widehat T:B_{X^{**}}\rightarrow Y^{**}\), a generalization of results of Arens [loc. cit.] and Aron-Berner [loc. cit.]. The tool used here is an infinite-dimensional converse Taylor theorem, which deserves well mentioning. Another main tool is a result of Zippin's style: The authors show that any weakly Cauchy sequence in \(X:=C(K)\) can be uniformly well approximated by a weakly Cauchy sequence from a certain subspace of \(X^{**}\), which is isometric to \(C[0,\alpha]\), \(\alpha\) a countable ordinal. Moreover, it is shown that, if a Banach space \(X\) does not contain a copy of \(\ell_1\), then the weak (respectively, uniformly weak) continuity of \(T:B_X\rightarrow Y\) is a sequential property in the sense that it is equivalent to \(T\) sending weakly convergent (respectively, weakly Cauchy) sequences from \(B_{X}\) into norm-convergent ones. This very interesting paper also includes a review of the most significant previous results in extensions of multilinear and polynomial mappings as well as a thorough presentation of the main concepts and properties needed in this context.
scientific article; zbMATH DE number 6135756
scientific article