Some Geometric Properties In Linear 2-normed Spaces

Linear 2-normed space is closely related to Banach space.Linear 2-normed space is a generalization of linear normed space,which is of great significance in control theory,approximation theory and other applications.The product of vector product,quantity product and the product of the norm we know well are special2-norms.GAHLER puts forward the concepts of 2-metric space,2-normed space and2-internal product space in 1962,which makes people gradually start to notice the generalized metric space and opens up people's exploration of generalized metric.Since 2-normed space is closely related to Banach space,many scholars begin to consider whether the properties or principles of Banach space can be extended to2-normed space.The main contents of this paper are the basic properties of 2-normed space defined by norm product and the geometric properties of linear 2-normed space with strict convexity and uniformly convexity,which are mainly divided into the following two parts:This paper first discusses a special 2-norm and gets rid of the positive definiteness of norm defined by GAHLER,which leads to the corresponding linear2-normed space.The properties of Banach space theory,especially Hilbert space theory and this 2-norm are applied to study the norm-dependent convergence,Hahn-Banach theorem,convex set separation theorem and compression mapping principle in this special 2-normed space defined by norm product.Riesz representation theorem is also extended to linear 2-normed space.Secondly,it is proved that compression mapping principle holds in general linear 2-normed space defined by GAHLER,and the fixed point set of nonexpansive mapping in strictly convex linear 2-normed space is convex.And this paper discusses the strict convexity and uniformly convexity of the finite dimensional linear2-normed space,and prove that the finite dimensional 2-normed space induced by the vector product is strict convex and thus uniformly convex.