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The Lindelöf property in Banach spaces

B. Cascales, I. Namioka, J. Orihuela (2003)

Studia Mathematica

A topological space (T,τ) is said to be fragmented by a metric d on T if each non-empty subset of T has non-empty relatively open subsets of arbitrarily small d-diameter. The basic theorem of the present paper is the following. Let (M,ϱ) be a metric space with ϱ bounded and let D be an arbitrary index set. Then for a compact subset K of the product space M D the following four conditions are equivalent: (i) K is fragmented by d D , where, for each S ⊂ D, d S ( x , y ) = s u p ϱ ( x ( t ) , y ( t ) ) : t ∈ S . (ii) For each countable subset A of D, ( K , d A ) is...

The Mackey-Arens theorem for non-locally convex spaces.

Jerzy Kakol (1990)

Collectanea Mathematica

Let R be a subcategory of the category of all topological vector spaces. Let E be an element of R. The problem of the existence of the finest R-topology on E with the same continuous linear functionals as the original one is discussed. Remarks concerning the Hahn-Banach Extension Property are included.

The measure extension problem for vector lattices

J. D. Maitland Wright (1971)

Annales de l'institut Fourier

Let V be a boundedly σ -complete vector lattice. If each V -valued premeasure on an arbitrary field of subsets of an arbitrary set can be extended to a σ -additive measure on the generated σ -field then V is said to have the measure extension property. Various sufficient conditions on V which ensure that it has this property are known. But a complete characterisation of the property, that is, necessary and sufficient conditions, is obtained here. One of the most useful characterisations is: V has the...

The non-archimedian space BC(X) with the strict topology.

Nicole De Grande-De Kimpe, Samuel Navarro (1994)

Publicacions Matemàtiques

Let X be a zero-dimensional, Hausdorff topological space and K a field with non-trivial, non-archimedean valuation under which it is complete. Then BC(X) is the vector space of the bounded continuous functions from X to K. We obtain necessary and sufficient conditions for BC(X), equipped with the strict topology, to be of countable type and to be nuclear in the non-archimedean sense.

The Oka-Weil theorem in topological vector spaces

Bui Dac Tac (1991)

Annales Polonici Mathematici

It is shown that a sequentially complete topological vector space X with a compact Schauder basis has WSPAP (see Definition 2) if and only if X has a pseudo-homogeneous norm bounded on every compact subset of X.

The order σ -complete vector lattice of AM-compact operators

Belmesnaoui Aqzzouz, Redouane Nouira (2009)

Czechoslovak Mathematical Journal

We establish necessary and sufficient conditions under which the linear span of positive AM-compact operators (in the sense of Fremlin) from a Banach lattice E into a Banach lattice F is an order σ -complete vector lattice.

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