Weak sequential completeness of sequence spaces.

Swartz, Charles. "Weak sequential completeness of sequence spaces.." Collectanea Mathematica 43.1 (1992): 55-61. <http://eudml.org/doc/41848>.

@article{Swartz1992,
abstract = {Köthe and Toeplitz introduced the theory of sequence spaces and established many of the basic properties of sequence spaces by using methods of classical analysis. Later many of these same properties of sequence spaces were reestablished by using soft proofs of functional analysis. In this note we would like to point out that an improved version of a classical lemma of Schur due to Hahn can be used to give very short proofs of two of the weak sequential completeness results of Köthe and Toeplitz. One of our proofs actually gives an improvement of one of the completeness results of Köthe and Toeplitz which was obtained by Benett using functional analysis methods and the method of proof is used in paragraph 3 to obtain a completeness result for ß-duals of vector-valued sequence spaces. One of our completeness results is employed to obtain a more general form of a Hellinger-Toeplitz type theorem for sequence spaces due to Köthe and the second completeness result is employed to obtain another Hellinger-Toeplitz type theorem for sequence spaces which covers additional cases not covered by Köthe's result.},
author = {Swartz, Charles},
journal = {Collectanea Mathematica},
keywords = {Completitud; Espacios de Banach; Análisis secuencial; Espacios vectoriales topológicos; theorem of Bennett and Kalton; -dual of a monotone sequence space is weakly sequentially complete; vector-valued sequence spaces; Hellinger-Toeplitz Theorems},
language = {eng},
number = {1},
pages = {55-61},
title = {Weak sequential completeness of sequence spaces.},
url = {http://eudml.org/doc/41848},
volume = {43},
year = {1992},
}

TY - JOUR
AU - Swartz, Charles
TI - Weak sequential completeness of sequence spaces.
JO - Collectanea Mathematica
PY - 1992
VL - 43
IS - 1
SP - 55
EP - 61
AB - Köthe and Toeplitz introduced the theory of sequence spaces and established many of the basic properties of sequence spaces by using methods of classical analysis. Later many of these same properties of sequence spaces were reestablished by using soft proofs of functional analysis. In this note we would like to point out that an improved version of a classical lemma of Schur due to Hahn can be used to give very short proofs of two of the weak sequential completeness results of Köthe and Toeplitz. One of our proofs actually gives an improvement of one of the completeness results of Köthe and Toeplitz which was obtained by Benett using functional analysis methods and the method of proof is used in paragraph 3 to obtain a completeness result for ß-duals of vector-valued sequence spaces. One of our completeness results is employed to obtain a more general form of a Hellinger-Toeplitz type theorem for sequence spaces due to Köthe and the second completeness result is employed to obtain another Hellinger-Toeplitz type theorem for sequence spaces which covers additional cases not covered by Köthe's result.
LA - eng
KW - Completitud; Espacios de Banach; Análisis secuencial; Espacios vectoriales topológicos; theorem of Bennett and Kalton; -dual of a monotone sequence space is weakly sequentially complete; vector-valued sequence spaces; Hellinger-Toeplitz Theorems
UR - http://eudml.org/doc/41848
ER -