### A generalization of the Banach-Steinhaus theorem for finite part limits

Ricardo Estrada and Jasson Vindas Diaz UGent (2017) 40(2). p.907-918
abstract
It is well known, as follows from the Banach-Steinhaus theorem, that if a sequence $\left\{ y_{n}\right\} _{n=1}^{\infty}$ of linear continuous functionals in a Fr\'{e}chet space converges pointwise to a linear functional $Y,$ $Y\left( x\right) =\lim_{n\rightarrow\infty}\left\langle y_{n} ,x\right\rangle$ for all $x,$ then $Y$ is actually continuous. In this article we prove that in a Fr\'{e}chet space\ the continuity of $Y$ still holds if $Y$ is the \emph{finite part} of the limit of $\left\langle y_{n},x\right\rangle$ as $n\rightarrow\infty.$ We also show that the continuity of finite part limits holds for other classes of topological vector spaces, such as \textsl{LF}-spaces, \textsl{DFS}-spaces, and \textsl{DFS} $^{\ast}$-spaces,\ and give examples where it does not hold.
author
organization
year
type
journalArticle (original)
publication status
published
subject
keyword
Finite part limits, Hadamard finite part, Banach-Steinhaus theorem
journal title
BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY
Bull. Malays. Math. Sci. Soc.
volume
40
issue
2
pages
907 - 918
Web of Science type
Article
Web of Science id
000406498300024
ISSN
0126-6705
2180-4206
DOI
10.1007/s40840-017-0450-7
language
English
UGent publication?
yes
classification
A1
I have transferred the copyright for this publication to the publisher
id
8518243
handle
http://hdl.handle.net/1854/LU-8518243
date created
2017-04-20 14:12:24
date last changed
2017-09-01 11:27:22
@article{8518243,
abstract     = {It is well known, as follows from the Banach-Steinhaus theorem, that if a sequence \${\textbackslash}left{\textbackslash}\{ y\_\{n\}{\textbackslash}right{\textbackslash}\} \_\{n=1\}\^{ }\{{\textbackslash}infty\}\$ of linear continuous functionals in a Fr{\textbackslash}'\{e\}chet space converges pointwise to a linear functional \$Y,\$ \$Y{\textbackslash}left( x{\textbackslash}right) ={\textbackslash}lim\_\{n{\textbackslash}rightarrow{\textbackslash}infty\}{\textbackslash}left{\textbackslash}langle y\_\{n\} ,x{\textbackslash}right{\textbackslash}rangle \$ for all \$x,\$ then \$Y\$ is actually continuous. In this article we prove that in a Fr{\textbackslash}'\{e\}chet space{\textbackslash} the continuity of \$Y\$ still holds if \$Y\$ is the {\textbackslash}emph\{finite part\} of the limit of \${\textbackslash}left{\textbackslash}langle y\_\{n\},x{\textbackslash}right{\textbackslash}rangle \$ as \$n{\textbackslash}rightarrow{\textbackslash}infty.\$ We also show that the continuity of finite part limits holds for other classes of topological vector spaces, such as {\textbackslash}textsl\{LF\}-spaces, {\textbackslash}textsl\{DFS\}-spaces, and {\textbackslash}textsl\{DFS\} \$\^{ }\{{\textbackslash}ast\}\$-spaces,{\textbackslash} and give examples where it does not hold.},
author       = {Estrada, Ricardo and Vindas Diaz, Jasson},
issn         = {0126-6705},
journal      = {BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY},
keyword      = {Finite part limits,Hadamard finite part,Banach-Steinhaus theorem},
language     = {eng},
number       = {2},
pages        = {907--918},
title        = {A generalization of the Banach-Steinhaus theorem for finite part limits},
url          = {http://dx.doi.org/10.1007/s40840-017-0450-7},
volume       = {40},
year         = {2017},
}


Chicago
Estrada, Ricardo, and Jasson Vindas Diaz. 2017. “A Generalization of the Banach-Steinhaus Theorem for Finite Part Limits.” Bulletin of the Malaysian Mathematical Sciences Society 40 (2): 907–918.
APA
Estrada, R., & Vindas Diaz, J. (2017). A generalization of the Banach-Steinhaus theorem for finite part limits. BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY, 40(2), 907–918.
Vancouver
1.
Estrada R, Vindas Diaz J. A generalization of the Banach-Steinhaus theorem for finite part limits. BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY. 2017;40(2):907–18.
MLA
Estrada, Ricardo, and Jasson Vindas Diaz. “A Generalization of the Banach-Steinhaus Theorem for Finite Part Limits.” BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY 40.2 (2017): 907–918. Print.