Complete Sets In Topological Linear Spaces

Since Meissner gave the notion of complete set for the first time in 1911,the problem of complete sets and their properties has received extensive attention from scholars.The study of complete sets and theri properties not only has important significance in practical application,but also has some challenges on the theoretical research.The paper is mainly concerned about the study of complete sets with respect to a given convex body and its properties in(real)topological linear spaces.First of all,notions of circumradius and diameter of bounded sets with respect to a given convex body in topological linear spaces are given.Based on notions of circumradius and diameter of bounded sets with respect to a given convex body,some results about circumradius and diameter of bounded sets with respect to a given convex body are discussed.Secondly,based on notions of wide spherical hull,tight spherical hull,adjoint transform and radius function of bounded sets with respect to a given convex body,some properties of wide spherical hull,tight spherical hull,adjoint transform and radius function of bounded sets with respect to a given convex body are discussed.And based on these,sufficient condition as well as some necessary and sufficient conditions for bounded sets with respect to a given convex body to be complete and conditions for bounded sets with respect to a given convex body to have a unique completion are provided.Thirdly,based on the notion of(K,u)-completeness,which was defined by J.P.Moreno and R.Schneider,the notion of(K,u)-completeness with respect to a given convex body is extended,and the relation between the notion of(K,u)-completeness and the uniqueness of completion with respect to a given convex body in topological linear spaces is discussed.Finally,based on Papini and Sen-lin Wu's construction of complete sets in Banach spaces,the construction of complete sets with respect to a given convex body in topological linear spaces is given.Moreover,based on Eggleston's construction of complete sets,the construction of complete sets in finite dimensional Banach spaces is given.