Previous |  Up |  Next


barrelled spaces; generalized sequences
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.
[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 10.1007/BF01194241 | MR 1193096
[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 10.1007/BF03322300 | MR 1215213
[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 10.1017/S0017089500030548 | MR 1260818
[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 1150746
[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 1300741
[FPS] M. Florencio, P. J. Paúl, C. Sáez: Barrelledness in $\lambda $-sums of normed spaces. Simon Stevin 63 (1989), 209–217. MR 1061568
[J] H. Jarchow: Locally Convex Spaces. B.G. Teubner. Stuttgart, 1981. MR 0632257 | Zbl 0466.46001
[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 1264704
[K] G. Köthe: Topological Vector Spaces I. Springer-Verlag, Berlin, Heidelberg and New York, 1969. MR 0248498
[L] P. Lurje: Tonnelierheit in lokalkonvexen Vektorgruppen. Manuscripta Math. 14 (1974), 107–121. DOI 10.1007/BF01171437 | MR 0367612
[BP] P. Pérez Carreras, J. Bonet: Barrelled Locally Convex Spaces. North-Holland Math. Studies, North-Holland, Amsterdam, New York, Oxford and Tokyo, 1987. MR 0880207
[R] R. C. Rosier: Dual spaces of certain vector sequence spaces. Pacific J. Math. 46 (1973), . DOI 10.2140/pjm.1973.46.487 | MR 0328544 | Zbl 0263.46009
Partner of
EuDML logo