$$ \int_{a}^{b} f^{2}(x) , \mathrm{d}x\int_{a}^{b} g^{2}(x) , \mathrm{d}x \geq \left(\int_{a}^{b} f(x)g(x) , \mathrm{d}x \right)^{2} $$
证明
$$ \begin{align} \int_{a}^{b} \left(f(x)+\lambda g(x)\right)^{2} , \mathrm{d}x & =\int_{a}^{b} \lambda^{2}g^{2}(x)+2f(x)g(x)\lambda+f^{2}(x) , \mathrm{d}x \ & =\left(\int_{a}^{b} g^{2}(x) , \mathrm{d}x \right)\lambda^{2}+2\left(\int_{a}^{b} f(x)g(x) , \mathrm{d}x \right)\lambda+\int_{a}^{b} f^{2}(x) , \mathrm{d}x \ & \geq 0 \end{align} $$
所以
$$ \Delta=4\left(\int_{a}^{b} f(x)g(x) , \mathrm{d}x \right)^{2}-4\int_{a}^{b} f^{2}(x) , \mathrm{d}x\int_{a}^{b} g^{2}(x) , \mathrm{d}x \leq 0 $$
得到 Schwarz 不等式