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Nonlinear absolutely summing mappings
Author(s) -
Matos Mário C.
Publication year - 2003
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200310087
Subject(s) - mathematics , absolute continuity , banach space , characterization (materials science) , homogeneous , pure mathematics , absolute convergence , space (punctuation) , nonlinear system , discrete mathematics , mathematical analysis , combinatorics , physics , fourier series , linguistics , philosophy , quantum mechanics , optics
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)