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

Marek Balcerzak, Artur Wachowicz, and Władysław Wilczyński. "Multiplying balls in the space of continuous functions on [0,1]." Studia Mathematica 170.2 (2005): 203-209. <http://eudml.org/doc/284859>.

@article{MarekBalcerzak2005,
abstract = {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 $Φ^\{-1\}(E)$ is residual whenever E is residual in C.},
author = {Marek Balcerzak, Artur Wachowicz, Władysław Wilczyński},
journal = {Studia Mathematica},
keywords = {Banach algebra; multiplication; continuous function; polygonal function; residual set},
language = {eng},
number = {2},
pages = {203-209},
title = {Multiplying balls in the space of continuous functions on [0,1]},
url = {http://eudml.org/doc/284859},
volume = {170},
year = {2005},
}

TY - JOUR
AU - Marek Balcerzak
AU - Artur Wachowicz
AU - Władysław Wilczyński
TI - Multiplying balls in the space of continuous functions on [0,1]
JO - Studia Mathematica
PY - 2005
VL - 170
IS - 2
SP - 203
EP - 209
AB - 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 $Φ^{-1}(E)$ is residual whenever E is residual in C.
LA - eng
KW - Banach algebra; multiplication; continuous function; polygonal function; residual set
UR - http://eudml.org/doc/284859
ER -