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On B r -completeness

Manuel Valdivia (1975)

Annales de l'institut Fourier

In this paper it is proved that if { E n } n = 1 and { F n } n = 1 are two sequences of infinite-dimensional Banach spaces then H = n = 1 E n × n = 1 F n is not B r -complete. If { E n } n = 1 and { F n } n = 1 are also reflexive spaces there is on H a separated locally convex topology , coarser than the initial one, such that H [ ] is a bornological barrelled space which is not an inductive limit of Baire spaces. It is given also another results on B r -completeness and bornological spaces.

On certain barrelled normed spaces

Manuel Valdivia (1979)

Annales de l'institut Fourier

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.

On nonbornological barrelled spaces

Manuel Valdivia (1972)

Annales de l'institut Fourier

If E is the topological product of a non-countable family of barrelled spaces of non-nulle dimension, there exists an infinite number of non-bornological barrelled subspaces of E . The same result is obtained replacing “barrelled” by “quasi-barrelled”.

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