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Balancing vectors and convex bodies

Wojciech Banaszczyk (1993)

Studia Mathematica

Let U, V be two symmetric convex bodies in n and |U|, |V| their n-dimensional volumes. It is proved that there exist vectors u 1 , . . . , u n U such that, for each choice of signs ε 1 , . . . , ε n = ± 1 , one has ε 1 u 1 + . . . + ε n u n r V where r = ( 2 π e 2 ) - 1 / 2 n 1 / 2 ( | U | / | V | ) 1 / n . Hence it is deduced that if a metrizable locally convex space is not nuclear, then it contains a null sequence ( u n ) such that the series n = 1 ε n u π ( n ) is divergent for any choice of signs ε n = ± 1 and any permutation π of indices.

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...

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...

Bounded linear maps between (LF)-spaces.

Angela A. Albanese (2003)

RACSAM

Characterizations of pairs (E,F) of complete (LF)?spaces such that every continuous linear map from E to F maps a 0?neighbourhood of E into a bounded subset of F are given. The case of sequence (LF)?spaces is also considered. These results are similar to the ones due to D. Vogt in the case E and F are Fréchet spaces. The research continues work of J. Bonet, A. Galbis, S. Önal, T. Terzioglu and D. Vogt.

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