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Barrelledness of Generalized Sums of Normed Spaces

Authors:

Affiliation:

(1) Dpto Matemática Aplicada II, Escuela Superior Ingenieros, Camino de los Descubrimientos, s/n, 41092- Sevilla, Spain.;(2) Dpto Matemática Aplicada II, Escuela Superior Ingenieros, Camino de los Descubrimientos, s/n, 41092- Sevilla, Spain

Abstract:

Let (E_i)_iI be a family of normed spaces and lambda

a space of scalar generalized sequences. The lambda

-sum of the family (E_i)_iI of spaces is

$lambda left{ {left( {E_i } right)_{i in I} } right}: = left{ {left( {x_i } right)_{i in I} ,x_i in E_i ,{text{ and }}left( {left| {x_i } right|} right)_{i in I} in lambda } right}.$

Starting from the topology on lambda

and the norm topology on each E_i, a natural topology on lambda

{(E_i)_iI} can be defined. We give conditions for lambda

{(E_i)_iI} to be quasi-barrelled, barrelled or locally complete.

Keywords:

barrelled spaces generalized sequences

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