Problems Related To Normed Linear Spaces With Some Special Properties

Numbers of very great works have appeared on the research of characterizations of inner product spaces, meanwhile, many related important theories and spaces with some special properties has also appeared. In general normed linear spaces, these theories'application has not been fully studied and discussed, on the other hand, some geometrical properties and constants in the spaces with some special properties have not researched in depth.In this paper, some properties in normed linear spaces with property P_λ, which originated from the research process of inner product spaces'characterizations, are studied, some results concerning inner product spaces'characterizations are obtained based on some important theories in the geometry of Minkowski spaces in a different light.First, basic knowledge about inner product spaces and the developments of these spaces'characterizations research are introduced, the main results on these fields by several researchers are summarized, and, furthermore, preliminaries, background and significance of the content of this paper are presented.Second, uniform nonsquareness in normed linear spaces with property P_λare discussed, and improvement is made on some related theories about property P_λ. Meanwhile, some new characterizations of inner product spaces are showed differently based on property P_λand isosceles orthogonality. Moreover, some related definitions and basic results are obtained.Finally, a special normed linear space with property P_λis constructed, and its properties are presented on the careful research, including convexity, invariability under rotation, as well as the relation between isosceles orthogonality and Euclidean orthogonality on the unit sphere of this space. As an immediate corollary, the nonsquare constant of this special space is obtained, from which it indirectly shows that for a givenλthat is greater than zero, nonsquare constants of normed linear spaces with property P_λare not necessarily 2^1/2, moreover, normed linear spaces with property P_λare not necessarily strict convexity or uniform convexity. On the other hand, the equivalent relationships between isosceles orthogonality and Euclidean orthogonality on the unit spheres of spaces ( R²,‖·‖) with invariability under rotatingÏ€/2nare obtained, and values of nonsquare constants in this kind of spaces are also obtained for n is an even number.