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Bayoumi quasi-differential is not different from Fréchet-differential

Fernando Albiac, José Ansorena (2012)

Open Mathematics

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...

Bessaga's conjecture in unstable Köthe spaces and products

Zefer Nurlu, Jasser Sarsour (1993)

Studia Mathematica

Let F be a complemented subspace of a nuclear Fréchet space E. If E and F both have (absolute) bases ( e n ) resp. ( f n ) , then Bessaga conjectured (see [2] and for a more general form, also [8]) that there exists an isomorphism of F into E mapping f n to t n e π ( k n ) where ( t n ) is a scalar sequence, π is a permutation of ℕ and ( k n ) is a subsequence of ℕ. We prove that the conjecture holds if E is unstable, i.e. for some base of decreasing zero-neighborhoods ( U n ) consisting of absolutely convex sets one has ∃s ∀p ∃q ∀r l i m n ( d n + 1 ( U q , U p ) ) / ( d n ( U r , U s ) ) = 0 where...

Bidual Spaces and Reflexivity of Real Normed Spaces

Keiko Narita, Noboru Endou, Yasunari Shidama (2014)

Formalized Mathematics

In this article, we considered bidual spaces and reflexivity of real normed spaces. At first we proved some corollaries applying Hahn-Banach theorem and showed related theorems. In the second section, we proved the norm of dual spaces and defined the natural mapping, from real normed spaces to bidual spaces. We also proved some properties of this mapping. Next, we defined real normed space of R, real number spaces as real normed spaces and proved related theorems. We can regard linear functionals...

Biduality in (LF)-spaces.

Klaus D. Bierstedt, José Bonet (2001)

RACSAM

En la Sección 1 se pueban resultados abstractos sobre preduales y sobre bidualidad de espacios (LF). Sea E = indn En un espacio (LF), ponemos H = indn Hn para una sucesión de subespacios de Fréchet Hn de En con Hn ⊂ Hn+1. Investigamos bajo qué condiciones el espacio E es canónicamente (topológicamente isomorfo a) el bidual inductivo (H'b)'i o (incluso) al bidual fuerte de H. Los resultados abstractos se aplican en la Sección 2, especialmente a espacios (LF) ponderados de funciones holomorfas, pero...

Biequivalence vector spaces in the alternative set theory

Miroslav Šmíd, Pavol Zlatoš (1991)

Commentationes Mathematicae Universitatis Carolinae

As a counterpart to classical topological vector spaces in the alternative set theory, biequivalence vector spaces (over the field Q of all rational numbers) are introduced and their basic properties are listed. A methodological consequence opening a new view towards the relationship between the algebraic and topological dual is quoted. The existence of various types of valuations on a biequivalence vector space inducing its biequivalence is proved. Normability is characterized in terms of total...

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