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

• Autorzy: Marek Balcerzak , Artur Wachowicz , Władysław Wilczyński
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 46J10: Banach algebras of continuous functions, function algebras
• 54E52: Baire category, Baire spaces
• 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.)
• 46B25: Classical Banach spaces in the general theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2005
• Tom: 170
• Numer: 2
• Strony: 203-209

Daty

wydano

2005

Twórcy

autor

Marek Balcerzak

• Institute of Mathematics, Łódź Technical University, Wólczańska 215, 93-005 Łódź, Poland

autor

Artur Wachowicz

• Institute of Mathematics, Łódź Technical University, Wólczańska 215, 93-005 Łódź, Poland

autor

Władysław Wilczyński

• Faculty of Mathematics, University of Łódź, Banacha 22, 90-238 Łódź, Poland

Identyfikatory

DOI

10.4064/sm170-2-5