Nonlinear absolutely summing mappings

Abstract

Abstract A mapping f , defined on an open subset A of a Banach space E , with values in another Banach space F , such that ( f ( a + x j ) − f ( a )) ∞ j =1 is absolutely summable in F , whenever ( x j ) ∞ j =1 is unconditionally summable (respectively, absolutely summable) in E , is called absolutely summing (respectively, regularly summing) at the point a ∈ A . It is proved that f is regularly summing at a if, and only if, there are M > 0 and δ > 0, such that ‖ f ( a + x ) − f ( a )‖ ≤ M ‖ x ‖, for all ‖ x ‖ ≤ δ . This result has as a consequence a characterization of absolutely summing mappings by means of inequalities. This result is analogous to the well know characterization of the linear absolutely summing mappings. Several results and examples show that the existence of non–linear absolutely summing mappings is not a rare phenomena. A Dvoretzky–Rogers Theorem for n –homogeneous polynomials is proved. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)