The Completeness And Separability Of Operator Spaces And Isometric Theory Of (2, P)-Normed Spaces

In this paper, we first study operator spaces and affine operator spaces,obtain some important properties, then study the generalized isometric problem in linear(2, p )- normed spaces andn - normed spaces and separability ofβ- positive-homogeneous operator spaces.In chapter1, we study some properties of the operator spaces. First, when X is a normed space and Y is aβ- normed space, we get that there is a equivalent relation of the completeness between the continuous linear operator space B ( X , Y ) and the normed space Y . Second,we introduce affine and the bounded affine operator, and get that image space Y is complete when the bounded affine operator space BΤ( X , Y) is complete. At last, we obtain that both the normed space X and Y are separable when BΤ( X , Y) is separable.In chapter2, we study the generalized isometric problem in linear(2, p )- normed space. We combine with the relationship between the normed space and p -normed space and the characteristic of p -strictly convex to promote 2 -isometric to generalized 2 -isometric. That is, let E and F be two linear(2, p )- normed spaces, F is p -strictly convex. If there is an integer n0 > 1such that f :E→Fsatisfies (I) x - z , s - q≤1 - f ( x ) - f ( z ), f ( s ) - f ( q )≤x - z ,s - q; (II) x - z , s - q 1p = n0 - f ( x ) - f ( z ), f ( s ) - f ( q )1p≥n0. Then f ( x ) - f ( z ), f ( s ) - f ( q ) = x - z ,s - q for every x , z , s ,q∈Ewith s - q =α( y - z)or s - q =β( y - x ),α,β∈R, y∈E.In chapter3, we study the Aleksandrov problem in linearn - normed space. We combine with relationship between 2-normed space and n - normed space to promote generalized 2 - isometry to generalized n - isometry. That is, let E and F be two linear n - normed spaces, F is strictly convex. If there is an integer n0 > 1such that f :E→Fsatisfies (I) x1 - y1 , L , xn - yn≤1-f ( x1 ) - f ( y1 ), L , f ( xn ) - f ( y n )≤x1 - y1 , L ,xn -yn; (II) x1 - y1 , L , xn - y n= n0-f ( x1 ) - f ( y1 ), L , f ( xn ) - f ( y n)≥n0. Then f ( x1 ) - f ( y1 ), L , f ( xn ) - f ( y n ) = x1 - y1 , L ,xn -ynfor every x1 , L , xn , y1 , L ,y n∈E with xi - yi =α( z - y1)or xi - yi =β( z - x1),α,β∈R, z∈E,2≤i≤n.In chapter4, we study the separability problem in subadditiveβ- positive-homogeneous operator spaces. That is, when X is a normed space and Y is aβ- normed space, ( Bβ( X , Y), )is separable, then X is separable.