Barrelledness of generalized sums of normed spaces

Ariel Fernández; Miguel Florencio; J. Oliveros

Czechoslovak Mathematical Journal (2000)

  • Volume: 50, Issue: 3, page 459-465
  • ISSN: 0011-4642

Abstract

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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.

How to cite

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Fernández, Ariel, Florencio, Miguel, and Oliveros, J.. "Barrelledness of generalized sums of normed spaces." Czechoslovak Mathematical Journal 50.3 (2000): 459-465. <http://eudml.org/doc/30576>.

@article{Fernández2000,
abstract = {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.},
author = {Fernández, Ariel, Florencio, Miguel, Oliveros, J.},
journal = {Czechoslovak Mathematical Journal},
keywords = {barrelled spaces; generalized sequences; barrelled spaces; generalized sequences},
language = {eng},
number = {3},
pages = {459-465},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Barrelledness of generalized sums of normed spaces},
url = {http://eudml.org/doc/30576},
volume = {50},
year = {2000},
}

TY - JOUR
AU - Fernández, Ariel
AU - Florencio, Miguel
AU - Oliveros, J.
TI - Barrelledness of generalized sums of normed spaces
JO - Czechoslovak Mathematical Journal
PY - 2000
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 50
IS - 3
SP - 459
EP - 465
AB - 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.
LA - eng
KW - barrelled spaces; generalized sequences; barrelled spaces; generalized sequences
UR - http://eudml.org/doc/30576
ER -

References

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