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

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.

## marked as duplicate by Theo Bendit, Trevor Gunn, Lord Shark the Unknown, Karn Watcharasupat, onurcanbektasDec 20 '18 at 3:07

$$\int (f-ag) ^{2} \geq 0$$ 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.