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# The $\varepsilon_{\infty}$-product of a $b$-space by a quotient bornological space

### Abstract

We define the $\varepsilon_{\infty }$-product of a Banach space $G$\ by a quotient bornological space $E\mid F$ that we denote by $G\varepsilon _{\infty }(E\mid F)$, and we prove that $G$ is an $% \mathcal{L}_{\infty }$-space if and only if the quotient bornological spaces $G\varepsilon _{\infty }(E\mid F)$ and $% (G\varepsilon E)\mid (G\varepsilon F)$ are isomorphic. Also, we show that the functor $\mathbf{.\varepsilon }_{\infty }\mathbf{.}:\mathbf{Ban\times qBan\longrightarrow qBan}$ is left exact. Finally, we define the $\varepsilon _{\infty }$-product of a b-space by a quotient bornological space and we prove that if $G$ is an $% \varepsilon$b-space\ and $E\mid F$ is a quotient bornological space, then $(G\varepsilon E)\mid (G\varepsilon F)$ is isomorphic to $G\varepsilon _{\infty }(E\mid F)$.

### Article Information

 Title The $\varepsilon_{\infty}$-product of a $b$-space by a quotient bornological space Source Methods Funct. Anal. Topology, Vol. 13 (2007), no. 3, 211-222 MathSciNet MR2356755 Copyright The Author(s) 2007 (CC BY-SA)

### Authors Information

Belmesnaoui Aqzzouz
Universite Ibn Tofail, Faculte des Sciences, Departement de Mathematiques, Laboratoire d'Analyse Fonctionnelle, Harmonique et Complexe, B.P. 133, Kenitra, Morocco

### Citation Example

Belmesnaoui Aqzzouz, The $\varepsilon_{\infty}$-product of a $b$-space by a quotient bornological space, Methods Funct. Anal. Topology 13 (2007), no. 3, 211-222.

### BibTex

@article {MFAT379,
AUTHOR = {Aqzzouz, Belmesnaoui},
TITLE = {The $\varepsilon_{\infty}$-product of a $b$-space by a quotient bornological space},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {13},
YEAR = {2007},
NUMBER = {3},
PAGES = {211-222},
ISSN = {1029-3531},
URL = {http://mfat.imath.kiev.ua/article/?id=379},
}