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

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Balcerzak M., Wachowicz A., Wilczyński W.

Studia Mathematica

2005

tom 170

nr 2

203-209

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.

Ograniczanie wyników

1 Studia Mathematica

1 Balcerzak M.

1 Wachowicz A.

1 Wilczyński W.

1 2005

Wyniki wyszukiwania

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