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Bayoumi quasi-differential is not different from Fréchet-differential

100%

Albiac F., Ansorena J.

Open Mathematics

2012

tom 10

nr 3

1071-1075

Unlike for Banach spaces, the differentiability of functions between infinite-dimensional nonlocally convex spaces has not yet been properly studied or understood. In a paper published in this Journal in 2006, Bayoumi claimed to have discovered a new notion of derivative that was more suitable for all F-spaces including the locally convex ones with a wider potential in analysis and applied mathematics than the Fréchet derivative. The aim of this short note is to dispel this misconception, since it could hinder making headway in this already hard enough subject. To that end we show that Bayoumi quasi-differentiability, when properly defined, is the same as Fréchet differentiability, and that some of his alleged applications are wrong.

On isomorphisms of some Köthe function F-spaces

81%

Kholomenyuk V., Mykhaylyuk V., Popov M.

Open Mathematics

2011

tom 9

nr 6

1267-1275

We prove that if Köthe F-spaces X and Y on finite atomless measure spaces (ΩX; ΣX, µX) and (ΩY; ΣY; µY), respectively, with absolute continuous norms are isomorphic and have the property $\mathop {\lim }\limits_{\mu (A) \to 0} \left\| {\mu (A)^{ - 1} 1_A } \right\| = 0$ (for µ = µX and µ = µY, respectively) then the measure spaces (ΩX; ΣX; µX) and (ΩY; ΣY; µY) are isomorphic, up to some positive multiples. This theorem extends a result of A. Plichko and M. Popov concerning isomorphic classification of L p(µ)-spaces for 0 < p < 1. We also provide a new class of F-spaces having no nonzero separable quotient space.

Ograniczanie wyników

2 Open Mathematics

1 Albiac F.

1 Ansorena J.

1 Kholomenyuk V.

1 Mykhaylyuk V.

1 Popov M.

1 2012

1 2011

Wyniki wyszukiwania

Bayoumi quasi-differential is not different from Fréchet-differential

On isomorphisms of some Köthe function F-spaces