Ball-Covering Property Of Banach Space And Quotient Space

In 2006, Cheng[1] has studied the geometric property of Banach space in different view, He gave a new kind of seguence of results in this field. for instance: the sphere of the unit ball of every n—dimensional space can be covered by 2n—balls off the origin at least. if X is smooth and if dim X=n, then S_X can be covered by n+1—balls off the origin. if the sphere of the unit ball of a Banach space can be covered by a sequence of balls off the origin with radius less than 1,then the space is separable. Ball—covering property are not homeomorphism invariant;atc.In this paper, We further studied that if X is a finite dimensional Banach space with dim X=n, if the set E of all w*—exposed points of Bx* is not Completely Contained in finitely many hyperplanes of X*, then S_X admits a ball—covering consisting of n+1 balls; if X is an infinite dimensional Banach space,then there exist an equivalent norm︱·︱on X and a closed subspace Y of X with dim X/Y=∞, X/Y has the ball—covering property with respect to︱·︱; A Banach space X admits an infinite dimensional separable quotient space if and only if it admits an infinite dimensional quotient space satisfying the unit sphere of the quotient space has a countable ball—covering with radii at most r for some r<1...