Knopp’s core theorem and subsequences of a bounded sequence

Abstract

Let sn, be a sequence in a p-dimensional Euclidean space EP. Let Kn=K(sn, Sn+l, ) be the convex hull of Sn Sn+l, and Kn its closure. The core of Sn is defined as n ln =Kn. Knopp's core theorem states that if A = (aij) is an infinite regular mnatrix with nonnegative elements, then the core of the A-transform of Sn is contained in the core of sn. In particular if Sn is bounded, every A-limit of a subsequence of sn is contained in the convex hull of limit points of Sn. With certain restrictions on A, the converse is also true; i.e., for any element t in the convex hull of limit points of sn, there is a subsequence of Sn which is A-limitable to t. The main objective of this paper is to show that for any t in the convex hull of limit points of a bounded sequence Sn, there is a subsequence of sn which is C1and E1-limitable to t. The following is Knopp's core theorem in EP.