Weakly sequentially continuous differentiable mappings |
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Authors: | Raffaella Cilia Joaquín M Gutirrez |
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Institution: | aDipartimento di Matematica, Facoltà di Scienze, Università di Catania, Viale Andrea Doria 6, 95125 Catania, Italy;bDepartamento de Matemática Aplicada, ETS de Ingenieros Industriales, Universidad Politécnica de Madrid, C. José Gutiérrez Abascal 2, 28006 Madrid, Spain |
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Abstract: | It is well known that a (linear) operator between Banach spaces is completely continuous if and only if its adjoint 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 UX is an open convex subset, let be a differentiable mapping whose derivative 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 . 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 ℓ1 and of Banach spaces without the Schur property containing a copy of ℓ1. 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 as above is locally weakly sequentially continuous. |
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Keywords: | Differentiable mapping Derivative Weakly sequentially continuous mapping Completely continuous mapping Banach space not containing color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6WK2-4WNXV4N-6&_mathId=mml13&_user=10&_cdi=6894&_rdoc=28&_acct=C000069468&_version=1&_userid=6189383&md5=290263a09abbd0ff5ce3fcc062d40dfe" title="Click to view the MathML source" ℓ" target="_blank">alt="Click to view the MathML source">ℓ 1 Hereditary Dunford– Pettis property |
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