Multiplying balls in the space of continuous functions on $[0,1]$

Marek Balcerzak ¹   ; Artur Wachowicz ¹   ; Władysław Wilczyński ²

¹ Institute of Mathematics Łódź Technical University Wólczańska 215 93-005 Łódź, Poland
² Faculty of Mathematics University of Łódź Banacha 22 90-238 Łódź, Poland

Studia Mathematica, Tome 170 (2005) no. 2, pp. 203-209

Résumé

Let $C$ denote the Banach space of real-valued continuous functions on $[0,1]$. Let $\Phi\colon C\times C\to C$. If $\Phi\in \{ +,\min ,\max\}$ then $\Phi$ is an open mapping but the multiplication $\Phi =\cdot$ is not open. For an open ball $B(f,r)$ in $C$ let $B^2(f,r)=B(f,r)\cdot B(f,r)$. Then $ f^2\in\mathop{\rm Int} B^2(f,r)$ for all $r>0$ if and only if either $f\ge 0$ on $[0,1]$ or $f\le 0$ on $[0,1]$. Another result states that $\mathop{\rm Int}(B_1\cdot B_2)\neq\emptyset$ for any two balls $B_1$ and $B_2$ in $C$. We also prove that if $\Phi\in\{+,\cdot,\min,\max\}$, then the set $\Phi^{-1}(E)$ is residual whenever $E$ is residual in $C$.

Détail

DOI : 10.4064/sm170-2-5

Keywords: denote banach space real valued continuous functions phi colon times phi min max phi mapping multiplication phi cdot ball cdot mathop int only either another result states mathop int cdot neq emptyset balls prove phi cdot min max set phi residual whenever residual