拟Banach空间正交的右存在性和左存在性

Right-Existence and Left-Existence of Orthogonality in Quasi-Banach Spaces

给出了一类新的正交性—拟Banach空间正交性,它是正交性的一种推广。首先,建立了拟Banach空间中两个元素的正交性与线性泛函之间的关系,并给出拟Banach空间正交的充要条件,即设X是实数域R上的拟Banach空间,有界线性泛函f∈SX*=f∈X*:‖f‖=1{},非零元素x∈X,H={h∈H:f(h)=0}是X的超平面,则f(x)=‖x‖等价于x⊥H;然后,给出了拟Banach空间正交右存在性和左存在性的充分条件;最后,举例说明了拟Banach空间中任意两元素不一定有正交右存在性。

A class of new generalized orthogonality of quasi-Banach spaces is given in this paper.It is a generalization of orthogonality.First,the relation of orthogonality and linear functional are introduced.At the same time,a necessary and sufficient condition of orthogonality in quasi-Banach spaces is given.Namely,let X be a quasi-Banach spaces on R,f∈SX＊=f∈X＊∶‖f‖=1 for all x∈X and x≠0,H={h∈H∶f（h）=0},then fx=‖x‖ is equivalent to x⊥H.The sufficient conditions of right-existence and left-existence of orthogonality are discussed.Finally,two examples which show that right-existence of orthogonality in quasi-Banach spaces does not exist for all elements are given.