Products of n open subsets in the space of continuous functions on [0,1]

• Autorzy: Ehrhard Behrends
• Języki publikacji: EN

Abstrakty

EN

Let O₁,...,Oₙ be open sets in C[0,1], the space of real-valued continuous functions on [0,1]. The product O₁ ⋯ Oₙ will in general not be open, and in order to understand when this can happen we study the following problem: given f₁,..., fₙ ∈ C[0,1], when is it true that f₁ ⋯ fₙ lies in the interior of $B_{ε}(f₁) ⋯ B_{ε}(fₙ)$ for all ε > 0 ? ($B_{ε}$ denotes the closed ball with radius ε and centre f.) The main result of this paper is a characterization in terms of the walk t ↦ γ(t): = (f₁(t),..., fₙ(t)) in ℝⁿ. It has to behave in a certain admissible way when approaching {x ∈ ℝⁿ | x₁ ⋯ xₙ = 0}. We will also show that in the case of complex-valued continuous functions on [0,1] products of open subsets are always open

Kategorie tematyczne

• 46E15: Banach spaces of continuous, differentiable or analytic functions
• 46B20: Geometry and structure of normed linear spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2011
• Tom: 204
• Numer: 1
• Strony: 73-95

Daty

wydano

2011

Twórcy

autor

Ehrhard Behrends

• Mathematisches Institut, Freie Universität Berlin, Arnimallee 6, D-14195 Berlin, Germany

Identyfikatory

DOI

10.4064/sm204-1-5