Stability of the ball-covering property


A normed space X is said to have the ball-covering property (BCP, for short) if its unit sphere can be covered by the union of countably many closed balls not containing the origin. Let (Ω, Σ, µ) be a separable measure space and X be a normed space. We show that Lp(µ, X) (1 ≤ p < ∞) has the BCP if and only if X has the BCP. We also prove that if {Xk} is a sequence of normed spaces, then X = (Σ ⊕ Xk)p has the BCP if and only each Xk has the BCP, where 1 ≤ p ≤ ∞. However, it is shown that L∞[0, 1] fails the BCP.

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Studia Mathematica