【摘要】：假設(shè)(33)是Banach空間X中由閉球(或開球)所構(gòu)成的球簇,如果每個球都不包含原點,并且所有球的并覆蓋了X的單位球面XS,則稱(33)是X的一個球覆蓋.如果X存在一個由可數(shù)多個球所構(gòu)成的球覆蓋,則稱X具有球覆蓋性質(zhì)(ball-covering property,簡寫為BCP).文獻(xiàn)]1[證明了:對于Gateaux可微空間(GDS)X和Y,它們具有BCP當(dāng)且僅當(dāng)它們的乘積空間(?),具有BCP,其中(?)本文沒有GDS的條件下,證明了,對于Banach空間X與Y,它們具有BCP當(dāng)且僅當(dāng)X×Y在范數(shù)(?)具有BCP,其中1≤p≤∞.其次,我們把有限乘積空間的BCP問題推廣到無限乘積,也就是說,如果X_k是具有BCP的Banach空間,則(?)也具有BCP,其中k∈N,1≤p≤∞。
[Abstract]:Suppose (33) is a cluster of closed balls (or tee balls) in Banach space X, if each ball does not contain the origin, and the unit sphere XS, of all balls and covering X is called (33) a ball cover of X. If X has a ball covering consisting of countable balls, then X has the property of ball covering (ball-covering property, abbreviated as BCP). [it is proved that for Gateaux differentiable spaces (GDS) X and Y, they have BCP if and only if their product spaces (?) and BCP, spaces (?) In this paper, we prove that for Banach spaces X and Y, they have BCP if and only if X 脳 Y is in the norm (?) We have BCP, where 1 鈮，

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