Comment on 'A uniform boundedness theorem for locally convex cones' [W. Roth, Proc. Amer. Math. Soc. 126 (1998), 1973-1982]

Abstract

In page 1975 of [W. Roth, A uniform boundedness theorem for locally convex cones, Proc. Amer. Math. Soc. 126 (1998), no.7, 1973-1982] we can see: In a locally convex vector space $E$ a barrel is defined to be an absolutely convex closed and absorbing subset $A$ of $E$. The set $U = \{(a,b)\in E^2,\ a-b\in A\}$ then is seen to be a barrel in the sense of Roth's definition. With a counterexample, we show that it is not enough for $U$ to be a barrel in the sense of Roth's definition. Then we correct this error with providing its converse and an application.

Key words: Locally convex cone, barrel.

Full Text

Article Information

Title	Comment on 'A uniform boundedness theorem for locally convex cones' [W. Roth, Proc. Amer. Math. Soc. 126 (1998), 1973-1982]
Source	Methods Funct. Anal. Topology, Vol. 20 (2014), no. 3, 292-295
MathSciNet	MR3242710
zbMATH	1324.46017
Milestones	Received 02/06/2013
Copyright	The Author(s) 2014 (CC BY-SA)

Authors Information

Davod Saeedi
Department of Mathematics, Payame Noor University, P.O. Box 19395-3697 Tehran, Iran.

Ismail Nikoufar
Department of Mathematics, Payame Noor University, P.O. Box 19395-3697 Tehran, Iran.

Husain Saiflu
Department of Pure Mathematics, Faculty of Mathematical Sciences, Tabriz University, Tabriz, Iran. 02/06/2013

Citation Example

Davod Saeedi, Ismail Nikoufar, and Husain Saiflu, Comment on 'A uniform boundedness theorem for locally convex cones' [W. Roth, Proc. Amer. Math. Soc. 126 (1998), 1973-1982], Methods Funct. Anal. Topology 20 (2014), no. 3, 292-295.

BibTex

@article {MFAT700,
    AUTHOR = {Saeedi, Davod and Nikoufar, Ismail and Saiflu, Husain},
     TITLE = {Comment on 'A uniform boundedness theorem for locally convex cones' [W. Roth, Proc. Amer. Math. Soc. 126 (1998), 1973-1982]},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {20},
      YEAR = {2014},
    NUMBER = {3},
     PAGES = {292-295},
      ISSN = {1029-3531},
  MRNUMBER = {MR3242710},
 ZBLNUMBER = {1324.46017},
       URL = {http://mfat.imath.kiev.ua/article/?id=700},
}