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Valdivia compacta and equivalent norms

Ondřej Kalenda (2000)

Studia Mathematica

We prove that the dual unit ball of a Banach space X is a Corson compactum provided that the dual unit ball with respect to every equivalent norm on X is a Valdivia compactum. As a corollary we show that the dual unit ball of a Banach space X of density ℵ 1 is Corson if (and only if) X has a projectional resolution of the identity with respect to every equivalent norm. These results answer questions asked by M. Fabian, G. Godefroy and V. Zizler and yield a converse to Amir-Lindenstrauss’ theorem.

Valuations of lines

Josef Mlček (1992)

Commentationes Mathematicae Universitatis Carolinae

We enlarge the problem of valuations of triads on so called lines. A line in an e -structure 𝔸 = 〈 A , F , E 〉 (it means that 〈 A , F 〉 is a semigroup and E is an automorphism or an antiautomorphism on 〈 A , F 〉 such that E ∘ E = 𝐈𝐝 ↾ A ) is, generally, a sequence 𝔸 ↾ B , 𝔸 ↾ U c , c ∈ 𝐅𝐙 (where 𝐅𝐙 is the class of finite integers) of substructures of 𝔸 such that B ⊆ U c ⊆ U d holds for each c ≤ d . We denote this line as 𝔸 ( U c , B ) c ∈ 𝐅𝐙 and we say that a mapping H is a valuation of the line 𝔸 ( U c , B ) c ∈ 𝐅𝐙 in a line 𝔸 ^ ( U ^ c , B ^ ) c ∈ 𝐅𝐙 if it is, for each c ∈ 𝐅𝐙 , a valuation of the triad 𝔸 ( U c , B ) in 𝔸 ^ ( U ^ c , B ^ ) . Some theorems on an existence of...

Variations of uniform completeness related to realcompactness

Miroslav Hušek (2017)

Commentationes Mathematicae Universitatis Carolinae

Various characterizations of realcompactness are transferred to uniform spaces giving non-equivalent concepts. Their properties, relations and characterizations are described in this paper. A Shirota-like characterization of certain uniform realcompactness proved by Garrido and Meroño for metrizable spaces is generalized to uniform spaces. The paper may be considered as a unifying survey of known results with some new results added.

Vietoris topology on spaces dominated by second countable ones

Carlos Islas, Daniel Jardon (2015)

Open Mathematics

For a given space X let C(X) be the family of all compact subsets of X. A space X is dominated by a space M if X has an M-ordered compact cover, this means that there exists a family F = FK : K ∈ C(M) ⊂ C(X) such that ∪ F = X and K ⊂ L implies that FK ⊂ FL for any K;L ∈ C(M). A space X is strongly dominated by a space M if there exists an M-ordered compact cover F such that for any compact K ⊂ X there is F ∈ F such that K ⊂ F . Let K(X) D C(X){Øbe the set of all nonempty compact subsets of a space...

Vitali sets.

Lahiri, Benoy Kumar, Lahiri, Indrajit (2001)

Bulletin of the Malaysian Mathematical Sciences Society. Second Series

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