Displaying similar documents to “Remarks on bounded sets in ( L F ) t v -spaces”

Regular inductive limits of K-spaces.

Thomas E. Gilsdorf (1991)

Collectanea Mathematica

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A well-known result for bounded sets in inductive limits of locally convex spaces is the following: If each of the constituent spaces En are Fréchet spaces and E is the inductive limit of the spaces En, then each bounded subset of E is bounded in some En iff E is locally complete. Using DeWilde's localization theorem, we show here that the completeness of each En and the local completeness of E may be replaced with the conditions that the spaces En are all webbed K-spaces and E is locally...

A revised closed graph theorem for quasi-Suslin spaces

Juan Carlos Ferrando, J. Kąkol, M. Lopez Pellicer (2009)

Czechoslovak Mathematical Journal

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Some results about the continuity of special linear maps between F -spaces recently obtained by Drewnowski have motivated us to revise a closed graph theorem for quasi-Suslin spaces due to Valdivia. We extend Valdivia’s theorem by showing that a linear map with closed graph from a Baire tvs into a tvs admitting a relatively countably compact resolution is continuous. This also applies to extend a result of De Wilde and Sunyach. A topological space X is said to have a (relatively countably)...

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.