Research On A Type Of Space Weaker Than Euclidean Space

A large number of fundamental results have been obtained by several researchers during their studying properties of generalized orthogonalities and the relationship beetween different kinds of generalized orthogonalities, providing basic theories for us. Their research mainly focus on the properties of generalized orthogonalities on the entire space and their impacts on the properties of the entire space, while research on the specific representations of two orthogonal elements mainly focus on symmetric Minkowski plane. Ji Donghai and Wu Senlin point out that isosceles orthogonality on the unit sphere of symmetric Minkowski plane is consistent with isosceles orthogonality in European space, but the research on the spaces with such consistency is not sufficient.Based on all that have mentioned above, first, beginning with generalized orthogonality in normed linear space, a new normed linear space--Weak Euclidean space is defined according to the consistency beetween isosceles orthogonality and European orthogonality in two-dimensional real normed linear space. In the light of Weak Euclidean space, in this paper, we study the properties of Weak Euclidean space and prove that Weak Euclidean space has pi/2-property. Meanwhile, it is proved that symmetric Minkowski plane is Weak Euclidean Space.Second, we discuss some common Weak Euclidean Spaces, for example: when axes of symmetry are (1,0) and (0,1), we consider invariability under rotation in Weak Euclidean Space and rotation characterization of the unit sphere while nonsquare constant equals to 2^1/2 . Besides, we present the expression of nonsquare constant in Weak Euclidean Space.Finally, we do more in-depth studies from the relationship between Weak Euclidean space and inner product space, and prove that strictly convex Weak Euclidean space is not necessary to be inner product space. At the same time we obtain a necessary and sufficient condition that Weak Euclidean spaces are inner product spaces. In addition, it is showed that the conjugate spaces of Weak Euclidean Spaces are still Weak Euclidean Spaces.