Multiplying balls in the space of continuous functions on $[0,1]$
Volume 170 / 2005
Studia Mathematica 170 (2005), 203-209
MSC: 46J10, 46B25, 26A15, 54E52.
DOI: 10.4064/sm170-2-5
Abstract
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 $C$.
