By a ball-coveringβ= {Br }(?) of a Banach space X, we mean that it is a collection of open balls off the origin whose union contains the sphere of the unit ball of X. The space X is said to have ball-covering property, if it admits a ball-covering consisting of countable many balls. A ball-coveringβis said to be minimal if the cardinal ofβis less than or equal to the cardinal of every ball-covering of X. Article [7] showed that if dimX = n, then n + 1≤βmin#≤2n ; if, in addition, X is smooth, thenβmin#=n+1;βmin#=2n if and only if X is isometric to (Rn,‖‖∞);If X is a Gateaux differentiability space or a locally uniformly convex space, then the unit sphere admits such a countable ball-covering if and only if X* is w* -separable. By a lemma of [8], this paper presents that ifβmin#=2n-1 , then there is an n-1 dimensional subspace of X isometric to (Rn-1,‖‖∞). Moreover, on this basis , this text combines article [11], first step's discussing the relation of the hereditarily indecomposable space and the ball-covering property space.
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