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 
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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/

[DFP] J. C. Díaz, M. Florencio, P. J. Paúl: A uniform boundedness theorem for $L^{\infty }(\mu ,E)$. Arch. Math. (Basel) 60 (1993), 73–78. | DOI | MR

[DFFP] S. Díaz, A. Fernández, M. Florencio, P. J. Paúl: An abstract Banach-Steinhaus theorem and applications to function spaces. Results Math.  23 (1993), 242–250. | DOI | MR

[DFP1] L. Drewnowski, M. Florencio, P. J. Paúl: Barrelled subspaces of spaces with subseries decompositions or Boolean rings of projections. Glasgow Math. J.  36 (1994), 57–69. | DOI | MR

[DFP2] L. Drewnowski, M. Florencio, P. J. Paúl: Barrelled function spaces. Progress in Functional Analysis, K.D. Bierstedt et al. (eds.), North-Holland Math. Studies, Elsevier/North-Holland, Amsterdam, Oxford, New York and Tokyo, 1992, pp. 191–199. | MR

[DFP3] L. Drewnowski, M. Florencio, P. J. Paúl: On the barrelledness of spaces of bounded vector functions. Arch. Math. (Basel) 63 (1994), 449–458. | MR

[FPS] M. Florencio, P. J. Paúl, C. Sáez: Barrelledness in $\lambda $-sums of normed spaces. Simon Stevin 63 (1989), 209–217. | MR

[J] H. Jarchow: Locally Convex Spaces. B.G. Teubner. Stuttgart, 1981. | MR | Zbl

[KR] J. Kakol, W. Roelcke: On the barrelledness of $\ell ^{p}$-direct sums of seminormed spaces for $1\le p\le \infty $. Arch. Math. (Basel) 62 (1994), 331–334. | MR

[K] G. Köthe: Topological Vector Spaces I. Springer-Verlag, Berlin, Heidelberg and New York, 1969. | MR

[L] P. Lurje: Tonnelierheit in lokalkonvexen Vektorgruppen. Manuscripta Math. 14 (1974), 107–121. | DOI | MR

[BP] P. Pérez Carreras, J. Bonet: Barrelled Locally Convex Spaces. North-Holland Math. Studies, North-Holland, Amsterdam, New York, Oxford and Tokyo, 1987. | MR

[R] R. C. Rosier: Dual spaces of certain vector sequence spaces. Pacific J. Math. 46 (1973), . | DOI | MR | Zbl