摘要
通过引入一个正定二次型 :λ(x ,y) =‖α‖2 x2 -2 (α ,β)xy +‖β‖2 y2 ,其中α和 β是内积空间E中的任意两个向量 ,x=(β ,γ) ,y=(α,γ)都是γ的泛函 ,γ∈E ,‖γ‖=1,建立了著名的Cauchy-Bunyzkowski-Schwarz(CBS)不等式的一个改进 ,提出了等式成立的新的见解 ,同时 ,分别建立了常用的Canchy不等式和Schwarz不等式的一个改进的具体表达式 ,给出了它的一些具体应用 ,并将改进后的Cauchy不等式推广到了Hilbert空间 ,而且对CBS不等式和改进后的CBS不等式进行了比较 ,对改进后的CBS不等式进行了评注。
Cauchy-Bunyakowski-Schwarz(CBS) inequality can be refined by means of the positive definite quadratic form λ(x,y) which is written as λ(x,y)=‖α‖ 2x 2-2(α,β)xy+‖β‖ 2y 2 ,where α and β are two elements in an inner product space E, x=(β,γ) and y=(α,γ) are functionals of γ,γ∈ E ,‖γ‖=1 and a new view which the equality holds is mentioned.At the same time,the forms of the refined Cauchy inequality and Schwarz inequalit are gotten,and some applications on CBS inequality are given,and the refined Cauchy inequality is extended with the CBS inequality and comments.
