Geodesic
Barrelledness of generalized sums of normed spaces
Czechoslovak Mathematical Journal, Tome 50 (2000) no. 3, pp. 459-465 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Let $(E_{i})_{i\in I}$ be a family of normed spaces and $\lambda $ a space of scalar generalized sequences. The $\lambda $-sum of the family $(E_{i})_{i\in I}$ of spaces is \[ \lambda \lbrace (E_{i})_{i\in I}\rbrace :=\lbrace (x_{i})_{i\in I},x_{i}\in E_{i}, \quad \text{and}\quad (\Vert x_{i}\Vert )_{i\in I}\in \lambda \rbrace . \] Starting from the topology on $\lambda $ and the norm topology on each $E_i,$ a natural topology on $\lambda \lbrace (E_i)_{i\in I}\rbrace $ can be defined. We give conditions for $\lambda \lbrace (E_i)_{i\in I}\rbrace $ to be quasi-barrelled, barrelled or locally complete.
Let $(E_{i})_{i\in I}$ be a family of normed spaces and $\lambda $ a space of scalar generalized sequences. The $\lambda $-sum of the family $(E_{i})_{i\in I}$ of spaces is \[ \lambda \lbrace (E_{i})_{i\in I}\rbrace :=\lbrace (x_{i})_{i\in I},x_{i}\in E_{i}, \quad \text{and}\quad (\Vert x_{i}\Vert )_{i\in I}\in \lambda \rbrace . \] Starting from the topology on $\lambda $ and the norm topology on each $E_i,$ a natural topology on $\lambda \lbrace (E_i)_{i\in I}\rbrace $ can be defined. We give conditions for $\lambda \lbrace (E_i)_{i\in I}\rbrace $ to be quasi-barrelled, barrelled or locally complete.
Classification : 46A08, 46A45, 46E10, 46E40
Keywords: barrelled spaces; generalized sequences 
@article{CMJ_2000_50_3_a1,
     author = {Fern\'andez, A. and Florencio, M. and Oliveros, J.},
     title = {Barrelledness of generalized sums of normed spaces},
     journal = {Czechoslovak Mathematical Journal},
     pages = {459--465},
     year = {2000},
     volume = {50},
     number = {3},
     mrnumber = {1777469},
     zbl = {1079.46500},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2000_50_3_a1/}
}
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Fernández, A.; Florencio, M.; Oliveros, J. Barrelledness of generalized sums of normed spaces. Czechoslovak Mathematical Journal, Tome 50 (2000) no. 3, pp. 459-465. http://geodesic.mathdoc.fr/item/CMJ_2000_50_3_a1/