Буняковский不等式之推广及其对于积分方程与希尔伯特空间之应用

<正> 不等式■(1) 通常称为布湼可夫斯基不等式,或席瓦耳智不等式,在本文中,作者推广此不等式为这里我们用 det u_(ij)(i,j=1,2,…,n)表第i列j行之元为 u_(ij)之n列行列式,f_i,g_j(i,j=1,2,…,n)表任一希尔伯特空间之任意二组之元,(f_i,g_j)表f_i与g_j二元之内乘积.

A GENERALIZATION OF BUNIAKOWSKY'S INEQUALITY WITH APPLICATIONS TO THE THEORY OF INTEGRAL EQUATIONS AND HILBERT SPACES

The inequality(?)(1)is usually called Schwarz's inequality,although it was stated first byBuniakowsky~1].In this paper,the author generalises(1)as follows|det(f~i,g_j)|~2≤det(f_i,f_j)det(g_i,g_j)(2)(i,j=1,2,...,n)where we use the notationdet u_(ij)(i,j=1,2,...,n)to denote the determinant of the n-th order of which the element in thei-th row and the j-th column is u_(ij)and,{f_i}and{g_i}(i=1,2,...,n)denote two sets of elements of an arbitrary Hilbert space,and(f_i,g_j)isthe inner product of f_i and g_j etc.Two special cases are that:i)for any two sets of L~2 functions{f_i(x)}and{g_i(x)}(i=1,2,...,n)defined in the interval α≤x≤b we have:(?)(i,j=1,2,...,n) (When n=1,(3)becomes(1)).ii)For any 2n sets of complex numbers{a_(ih)},{b_(hj)},(h=1,2,3,…;i,j=1,2,…,n),satisfying the conditions(?)we have(?)(4)(i,j=1,2,…,n)which is a generalization of the Cauchy's inequality(?)(5)By the aid of these inequalities,some new properties of integral equa-tions and Hilbert spaces are obtained.The main results are given in thefollowing theorems:Theorem 1.Let(?)be an abstract Hilbert space,and let K be a boundedpositive transformation on the space(?),then(?)(6)(i,j=1,2,…,n)for all elements f_i,g_j∈(?).Theorem 2.Let A(x,y)and B(x,y)be two arbitrary L~2 kernels,sothat(?)then for any positive integer n,we have:(?)(7)(?)(8)(i,i=1,2,…,n),where(?)(?)(?)(?)Applying(8)we obtain the inequality(?)(9) where{λ_hA]}(h=1,2,3,…)denotes the complete set of singular values(i.e.E.Schmidt characteristic values)of A(x,y)satisfying 0<λ_iA]≤≤λ_2A]≤….More accurately,we shall prove that(?)(h_11 we have:(?)(17)Further,if a function of several variables f(x_1,x_2,