Displaying similar documents to “On B r -completeness”

The space D ( U ) is not B r -complete

Manuel Valdivia (1977)

Annales de l'institut Fourier

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Certain classes of locally convex space having non complete separated quotients are studied and consequently results about B r -completeness are obtained. In particular the space of L. Schwartz D ( Ω ) is not B r -complete where Ω denotes a non-empty open set of the euclidean space R m .

Some characterizations of ultrabornological spaces

Manuel Valdivia (1974)

Annales de l'institut Fourier

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Let U be an infinite-dimensional separable Fréchet space with a topology defined by a family of norms. Let F be an infinite-dimensional Banach space. Then F is the inductive limit of a family of spaces equal to E . The choice of suitable classes of Fréchet spaces allows to give characterizations of ultrabornological spaces using the result above.. Let Ω be a non-empty open set in the euclidean n -dimensional space R n . Then F is the inductive limit of a family of spaces equal to D ( Ω ) . ...

Holomorphic functions on locally convex topological vector spaces. I. Locally convex topologies on ( U )

Sean Dineen (1973)

Annales de l'institut Fourier

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This article is devoted to a study of locally convex topologies on H ( U ) (where U is an open subset of the locally convex topological vector space E and H ( U ) is the set of all complex valued holomorphic functions on E ). We discuss the following topologies on H ( U ) : (a) the compact open topology I 0 , (b) the bornological topology associated with I 0 , (c) the ported topology of Nachbin I ω , (d) the bornological topology associated with I ω  ; and ...

A lifting theorem for locally convex subspaces of L 0

R. Faber (1995)

Studia Mathematica

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We prove that for every closed locally convex subspace E of L 0 and for any continuous linear operator T from L 0 to L 0 / E there is a continuous linear operator S from L 0 to L 0 such that T = QS where Q is the quotient map from L 0 to L 0 / E .

On the complemented subspaces of the Schreier spaces

I. Gasparis, D. Leung (2000)

Studia Mathematica

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It is shown that for every 1 ≤ ξ < ω, two subspaces of the Schreier space X ξ generated by subsequences ( e l n ξ ) and ( e m n ξ ) , respectively, of the natural Schauder basis ( e n ξ ) of X ξ are isomorphic if and only if ( e l n ξ ) and ( e m n ξ ) are equivalent. Further, X ξ admits a continuum of mutually incomparable complemented subspaces spanned by subsequences of ( e n ξ ) . It is also shown that there exists a complemented subspace spanned by a block basis of ( e n ξ ) , which is not isomorphic to a subspace generated by a subsequence of ( e n ζ ) ,...

Characterizations of elements of a double dual Banach space and their canonical reproductions

Vassiliki Farmaki (1993)

Studia Mathematica

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For every element x** in the double dual of a separable Banach space X there exists the sequence ( x ( 2 n ) ) of the canonical reproductions of x** in the even-order duals of X. In this paper we prove that every such sequence defines a spreading model for X. Using this result we characterize the elements of X**╲ X which belong to the class B 1 ( X ) B 1 / 2 ( X ) (resp. to the class B 1 / 4 ( X ) ) as the elements with the sequence ( x ( 2 n ) ) equivalent to the usual basis of 1 (resp. as the elements with the sequence ( x ( 4 n - 2 ) - x ( 4 n ) ) equivalent to the...

Mapping Properties of c 0

Paul Lewis (1999)

Colloquium Mathematicae

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Bessaga and Pełczyński showed that if c 0 embeds in the dual X * of a Banach space X, then 1 embeds as a complemented subspace of X. Pełczyński proved that every infinite-dimensional closed linear subspace of 1 contains a copy of 1 that is complemented in 1 . Later, Kadec and Pełczyński proved that every non-reflexive closed linear subspace of L 1 [ 0 , 1 ] contains a copy of 1 that is complemented in L 1 [ 0 , 1 ] . In this note a traditional sliding hump argument is used to establish a simple mapping property of...

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 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 0 ( X , 𝒜 ) covering it, there is a positive integer p such that E p is a dense barrelled subspace of 0 ( X , 𝒜 ) , and some new results in measure theory are deduced from this fact.