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Displaying similar documents to “A hereditary property in locally convex spaces”

Barrelledness of generalized sums of normed spaces

Ariel Fernández, Miguel Florencio, J. Oliveros (2000)

Czechoslovak Mathematical Journal

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Let ( E i ) i ∈ I be a family of normed spaces and λ a space of scalar generalized sequences. The λ -sum of the family ( E i ) i ∈ I of spaces is λ { ( E i ) i ∈ I } : = { ( x i ) i ∈ I , x i ∈ E i , and ( ∥ x i ∥ ) i ∈ I ∈ λ } . Starting from the topology on λ and the norm topology on each E i , a natural topology on λ { ( E i ) i ∈ I } can be defined. We give conditions for λ { ( E i ) i ∈ I } to be quasi-barrelled, barrelled or locally complete.

On certain barrelled normed spaces

Manuel Valdivia (1979)

Annales de l'institut Fourier

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Let 𝒜 be a σ -algebra on a set X . If A belongs to 𝒜 let e ( A ) be the characteristic function of A . Let ℓ 0 ∞ ( X , 𝒜 be the linear space generated by { e ( A ) : A ∈ 𝒜 } endowed with the topology of the uniform convergence. It is proved in this paper that if ( E n ) is an increasing sequence of subspaces of ℓ 0 ∞ ( X , 𝒜 ) covering it, there is a positive integer p such that E p is a dense barrelled subspace of ℓ 0 ∞ ( X , 𝒜 ) , and some new results in measure theory are deduced from this fact.

Generalized characterization of the convex envelope of a function

Fethi Kadhi (2002)

RAIRO - Operations Research - Recherche Opérationnelle

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We investigate the minima of functionals of the form ∫ [ a , b ] g ( u ˙ ( s ) ) d s where g is strictly convex. The admissible functions u : [ a , b ] → ℝ are not necessarily convex and satisfy u ≤ f on [ a , b ] , u ( a ) = f ( a ) , u ( b ) = f ( b ) , f is a fixed function on [ a , b ] . We show that the minimum is attained by f ¯ , the convex envelope of f .