Stability Of Ball-covering Property In Product Space

A family ? of (open) closed balls in a Banach space X is said to be a ball-covering of X if every ball in ? does not contain the origin in its interior and whose union covers the unit sphereXS of X;A Banach space is said to have the ball-covering property(BCP,in short)if its unit sphere can be contained in the union of countable many(open)closed balls off origin.Article[1] shows that Gateaux differentiable spaces(GDS)X and Y have BCP if and only if (?),have BCP,Where (?).But in thispaper,Without the GDS condition,we prove that Banach spaces X and Y have BCP if and only if (?),have BCP,where 1?p??.Secondly,we extend theball-covering of Product spaces of finite dimension to infinite dimension,i.e.if X_k is a Banach space with BCP,then (?) has BCP.where k?N,1?p??.