I'm self-studying Kolmogorov/Fomin's Introduction to Real Analysis and I'm stuck on a question. I'm required to prove Schwarz's inequality in $\displaystyle L^{2}$ by showing that

$\displaystyle \left(\int_{a}^{b}x(t)y(t)dt\right)^{2} = \int_{a}^{b}x^{2}(t)dt\int_{a}^{b}y^{2}(t)dt - \frac{1}{2}\int_{a}^{b}\int_{a}^{b}x(t)y(s)-x(s)y(t) ds dt $

I think I can do it by the limit definition of the integral and by proving that

$\displaystyle \left(\sum_{k=1}^{n} a_{k}b_{k}\right)^{2} = \sum_{k=1}^{n}a_{k}^{2}\sum_{k=1}^{n}b_{k}^{2} - \frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{n}(a_{i}b_{j}-a_{j}b_{i})$

Am I on the right track, or is there a better way that doesn't appeal to the definition of the integral?