Let C denote the Banach space of real-valued continuous functions on [0,1]. Let Φ: C × C → C. If Φ ∈ +, min, max then Φ is an open mapping but the multiplication Φ = · is not open. For an open ball B(f,r) in C let B²(f,r) = B(f,r)·B(f,r). Then f² ∈ Int B²(f,r) for all r > 0 if and only if either f ≥ 0 on [0,1] or f ≤ 0 on [0,1]. Another result states that Int(B₁·B₂) ≠ ∅ for any two balls B₁ and B₂ in C. We also prove that if Φ ∈ +,·,min,max, then the set is residual whenever E is residual in C.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm170-2-5,
author = {Marek Balcerzak and Artur Wachowicz and W\l adys\l aw Wilczy\'nski},
title = {Multiplying balls in the space of continuous functions on [0,1]},
journal = {Studia Mathematica},
volume = {166},
year = {2005},
pages = {203-209},
zbl = {1093.46023},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm170-2-5}
}
Marek Balcerzak; Artur Wachowicz; Władysław Wilczyński. Multiplying balls in the space of continuous functions on [0,1]. Studia Mathematica, Tome 166 (2005) pp. 203-209. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm170-2-5/