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Acknowledgement of priority: Separable quotients of Banach spaces.

Marek Wójtowicz (1998)

Collectanea Mathematica

In previous papers, it is proved, among other things, that every infinite dimensional sigma-Dedekind complete Banach lattice has a separable quotient. It has come to my attention that L. Weis proved this result without assuming sigma-Dedekind completeness; the proof is based, however, on the deep theorem of J. Hagler and W.B. Johnson concerning the structure of dual balls of Banach spaces and therefore cannot be applied simply to the case of locally convex solid topologically complete Riesz spaces....

Addendum to "Necessary condition for Kostyuchenko type systems to be a basis in Lebesgue spaces" (Colloq. Math. 127 (2012), 105-109)

Aydin Sh. Shukurov (2014)

Colloquium Mathematicae

It is well known that if φ(t) ≡ t, then the system φ ⁿ ( t ) n = 0 ∞ is not a Schauder basis in L₂[0,1]. It is natural to ask whether there is a function φ for which the power system φ ⁿ ( t ) n = 0 ∞ is a basis in some Lebesgue space L p . The aim of this short note is to show that the answer to this question is negative.

Adequate Compacta which are Gul’ko or Talagrand

Čížek, Petr, Fabian, Marián (2003)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 54H05, 03E15, 46B26We answer positively a question raised by S. Argyros: Given any coanalytic, nonalytic subset Σ′ of the irrationals, we construct, in Mercourakis space c1(Σ′), an adequate compact which is Gul’ko and not Talagrand. Further, given any Borel, non Fσ subset Σ′ of the irrationals, we construct, in c1(Σ′), an adequate compact which is Talagrand and not Eberlein.Supported by grants AV CR 101-90-03, and GA CR 201-01-1198

Affine bijections of C(X,I)

Janko Marovt (2006)

Studia Mathematica

Let 𝒳 be a compact Hausdorff space which satisfies the first axiom of countability, I = [0,1] and 𝓒(𝒳,I) the set of all continuous functions from 𝒳 to I. If φ: 𝓒(𝒳,I) → 𝓒(𝒳,I) is a bijective affine map then there exists a homeomorphism μ: 𝒳 → 𝒳 such that for every component C in 𝒳 we have either φ(f)(x) = f(μ(x)), f ∈ 𝓒(𝒳,I), x ∈ C, or φ(f)(x) = 1-f(μ(x)), f ∈ 𝓒(𝒳,I), x ∈ C.

Affine group acting on hyperspaces of compact convex subsets of ℝⁿ

Sergey A. Antonyan, Natalia Jonard-Pérez (2013)

Fundamenta Mathematicae

For every n ≥ 2, let cc(ℝⁿ) denote the hyperspace of all nonempty compact convex subsets of the Euclidean space ℝⁿ endowed with the Hausdorff metric topology. Let cb(ℝⁿ) be the subset of cc(ℝⁿ) consisting of all compact convex bodies. In this paper we discover several fundamental properties of the natural action of the affine group Aff(n) on cb(ℝⁿ). We prove that the space E(n) of all n-dimensional ellipsoids is an Aff(n)-equivariant retract of cb(ℝⁿ). This is applied to show that cb(ℝⁿ) is homeomorphic...

Algebraic and topological properties of some sets in ℓ₁

Taras Banakh, Artur Bartoszewicz, Szymon Głąb, Emilia Szymonik (2012)

Colloquium Mathematicae

For a sequence x ∈ ℓ₁∖c₀₀, one can consider the set E(x) of all subsums of the series ∑ n = 1 ∞ x ( n ) . Guthrie and Nymann proved that E(x) is one of the following types of sets: () a finite union of closed intervals; () homeomorphic to the Cantor set; homeomorphic to the set T of subsums of ∑ n = 1 ∞ b ( n ) where b(2n-1) = 3/4ⁿ and b(2n) = 2/4ⁿ. Denote by ℐ, and the sets of all sequences x ∈ ℓ₁∖c₀₀ such that E(x) has the property (ℐ), () and ( ), respectively. We show that ℐ and are strongly -algebrable and is -lineable. We...

Algunos resultados sobre sistemas de desigualdades lineales.

Juan Antonio Mira López (1988)

Trabajos de Investigación Operativa

En este artículo aplicamos la condición de Mazur-Orlicz para extender a espacios normados algunos resultados de consistencia de desigualdades lineales (s.d.l.) en Rn. Asimismo, obtenemos condiciones para la consistencia de s.d.l. en un espacio localmente convexo, cuando las soluciones pertenecen a ciertos subconjuntos del dual topológico.

Almost ball remotal subspaces in Banach spaces

Tanmoy Paul (2012)

Studia Mathematica

We investigate almost ball remotal and ball remotal subspaces of Banach spaces. Several subspaces of the classical Banach spaces are identified having these properties. Some stability results for these properties are also proved.

Almost demi Dunford--Pettis operators on Banach lattices

Hedi Benkhaled (2023)

Commentationes Mathematicae Universitatis Carolinae

We introduce new concept of almost demi Dunford–Pettis operators. Let E be a Banach lattice. An operator T from E into E is said to be almost demi Dunford–Pettis if, for every sequence { x n } in E + such that x n → 0 in σ ( E , E ' ) and ∥ x n - T x n ∥ → 0 as n → ∞ , we have ∥ x n ∥ → 0 as n → ∞ . In addition, we study some properties of this class of operators and its relationships with others known operators.

Almost exactness in normed spaces II

Robin Harte, Мostafa Мbekhta (1996)

Studia Mathematica

In the normed space of bounded operators between a pair of normed spaces, the set of operators which are "bounded below" forms the interior of the set of one-one operators. This note is concerned with the extension of this observation to certain spaces of pairs of operators.

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