Uniform Convergence and the Hahn-Schur Theorem

Let E be a vector space, F aset, G be a locally convex space, b : E X F - G a map such that ò(-,y): E - G is linear for every y G F; we write b(x, y) = x · y for brevity. Let ë be a scalar sequence space and w(E,F) the weakest topology on E such that the linear maps b(-,y): E - G are continuous for...

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Autor principal:	Swartz,Charles
Lenguaje:	English
Publicado:	Universidad Católica del Norte, Departamento de Matemáticas 2012
Materias:	Multiplier convergent series uniform convergence Hahn-Schur Theorem
Acceso en línea:	http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172012000200004
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spelling	oai:scielo:S0716-091720120002000042012-07-18Uniform Convergence and the Hahn-Schur TheoremSwartz,Charles Multiplier convergent series uniform convergence Hahn-Schur Theorem Let E be a vector space, F aset, G be a locally convex space, b : E X F - G a map such that ò(-,y): E - G is linear for every y G F; we write b(x, y) = x · y for brevity. Let ë be a scalar sequence space and w(E,F) the weakest topology on E such that the linear maps b(-,y): E - G are continuous for all y G F .A series Xj in X is ë multiplier convergent with respect to w(E, F) if for each t = {tj} G ë ,the series Xj=! tj Xj is w(E,F) convergent in E. For multiplier spaces ë satisfying certain gliding hump properties we establish the following uniform convergence result: Suppose j XX ij is ë multiplier convergent with respect to w(E, F) for each i G N and for each t G ë the set {Xj=! tj Xj : i} is uniformly bounded on any subset B C F such that {x · y : y G B} is bounded for x G E.Then for each t G ë the series ^jjLi tj xj · y converge uniformly for y G B,i G N. This result is used to prove a Hahn-Schur Theorem for series such that lim¿ Xj=! tj xj · y exists for t G ë,y G F. Applications of these abstract results are given to spaces of linear operators, vector spaces in duality, spaces of continuous functions and spaces with Schauder bases.info:eu-repo/semantics/openAccessUniversidad Católica del Norte, Departamento de MatemáticasProyecciones (Antofagasta) v.31 n.2 20122012-06-01text/htmlhttp://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172012000200004en10.4067/S0716-09172012000200004
institution	Scielo Chile
collection	Scielo Chile
language	English
topic	Multiplier convergent series uniform convergence Hahn-Schur Theorem
spellingShingle	Multiplier convergent series uniform convergence Hahn-Schur Theorem Swartz,Charles Uniform Convergence and the Hahn-Schur Theorem
description	Let E be a vector space, F aset, G be a locally convex space, b : E X F - G a map such that ò(-,y): E - G is linear for every y G F; we write b(x, y) = x · y for brevity. Let ë be a scalar sequence space and w(E,F) the weakest topology on E such that the linear maps b(-,y): E - G are continuous for all y G F .A series Xj in X is ë multiplier convergent with respect to w(E, F) if for each t = {tj} G ë ,the series Xj=! tj Xj is w(E,F) convergent in E. For multiplier spaces ë satisfying certain gliding hump properties we establish the following uniform convergence result: Suppose j XX ij is ë multiplier convergent with respect to w(E, F) for each i G N and for each t G ë the set {Xj=! tj Xj : i} is uniformly bounded on any subset B C F such that {x · y : y G B} is bounded for x G E.Then for each t G ë the series ^jjLi tj xj · y converge uniformly for y G B,i G N. This result is used to prove a Hahn-Schur Theorem for series such that lim¿ Xj=! tj xj · y exists for t G ë,y G F. Applications of these abstract results are given to spaces of linear operators, vector spaces in duality, spaces of continuous functions and spaces with Schauder bases.
author	Swartz,Charles
author_facet	Swartz,Charles
author_sort	Swartz,Charles
title	Uniform Convergence and the Hahn-Schur Theorem
title_short	Uniform Convergence and the Hahn-Schur Theorem
title_full	Uniform Convergence and the Hahn-Schur Theorem
title_fullStr	Uniform Convergence and the Hahn-Schur Theorem
title_full_unstemmed	Uniform Convergence and the Hahn-Schur Theorem
title_sort	uniform convergence and the hahn-schur theorem
publisher	Universidad Católica del Norte, Departamento de Matemáticas
publishDate	2012
url	http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172012000200004
work_keys_str_mv	AT swartzcharles uniformconvergenceandthehahnschurtheorem
_version_	1718439783289585664

Uniform Convergence and the Hahn-Schur Theorem

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