Cauchy Schwarz inequality for $\langle f,g\rangle = \int_0^1f(x)g(x)\mathrm{d}x$

Question

For the vector space of continuous functions on $[0,1]$ Define the inner product as $$\langle f,g\rangle = \int_0^1f(x)g(x)\mathrm{d}x$$ Please help me to prove the Cauchy Schwarz inequality for this given inner product.

Cauchy Schwarz Inequality: $|\langle v,u\rangle|\leq \lVert v\rVert\lVert u\rVert$ for the elements $v,u$ in the inner product space.

Answer 1

$\int (f-ag) ^{2} \geq 0$ so $\int f^{2}-2a\int fg +a^{2}\int g^{2} \geq 0$. Just put $a=\frac {\int fg} {\int g^{2}}$ and simplify.