Chauchy-Schwarze不等式

设f(x)和g(x)都在[a,b]上可积，则

$[∫_a^bf(x)g(x)dx]^2≤∫_a^bf^2(x)dx∫_a^bg^2(x)dx$

证明；对于任意的实数t，显然\([tf(x)+g(x)]^2在[a,b]上可积，且tf(x)+g(x)]^2≥0\)，则

$[∫_a^b[tf(x)+g(x)]^dx2≥0$

即

$t^2∫_a^bf^2(x)dx+2t∫_a^bf(x)g(x)dx+∫_a^bg^2(x)dx≥0                      $

则

$Δ=[2∫_a^bf(x)g(x)dx]^2-4∫_a^bf^2(x)dx∫_a^bg^2(x)dx≤0$

即

$[∫_a^bf(x)g(x)dx]^2≤∫_a^bf^2(x)dx∫_a^bg^2(x)dx$

标签：可积,2t,bf,bg,不等式,Chauchy,dx,tf,Schwarze
来源： https://www.cnblogs.com/valar-morghulis/p/14847504.html