赋范线性空间的次内积与次正交性

Second-Inner—Product and S—Orthogonality in Normed Linear Spaces

文章试图解决在一般的赋范线性空间中不能定义内积的问题。首先,在去掉内积可加性一条的较弱条件下,通过一正定齐次的Hermite泛函,引入了一般赋范线性空间的次内积,并进行了相应的讨论。然后,通过次内积引入了次正交性,并建立几个有趣的引理对次正交性的性质进行了必要的讨论。

It is well known that the usual inner-product cannot be defined in normed linear spaces and a useful notion for applications to non-linear functional analysis is that of orthogonality. Tho purpose of this paper is to discuss these problems.First of all, a notion that is called second-inner-product of normed linear spaces is introduced, i, e. a positive, homogeneous and symmetric binary Hermite functional that is said to be second-inner-product is defined, upon condition that eliminates additivity from 4 conditions of the inner-product on X x X. Through necessary discussion about this notion, some useful conclusions are given. Due to second-innor-product, the notion of second-orthogonality is naturally to be introduced. Furthermore, this notion is also considered and several properties are obtained.