Cauchy-Bunyakovsky-Schwarz Inequality/Definite Integrals

Theorem
Let $f$ and $g$ be real functions which are continuous on the closed interval $\left[{a \,.\,.\, b}\right]$.

Then:
 * $\displaystyle \left({\int_a^b f \left({t}\right) g \left({t}\right) \ \mathrm d t}\right)^2 \le \int_a^b \left({f \left({t}\right)}\right)^2 \mathrm d t \int_a^b \left({g \left({t}\right)}\right)^2 \mathrm d t$

Proof
where:
 * $\displaystyle A = \int_a^b \left({f \left({t}\right)}\right)^2 \mathrm d t$


 * $\displaystyle B = \int_a^b f \left({t}\right) g \left({t}\right) \ \mathrm d t$


 * $\displaystyle C = \int_a^b \left({g \left({t}\right)}\right)^2 \mathrm d t$

The quadratic equation $A x^2 + 2 B x + C$ is non-negative for all $x$.

It follows (using the same reasoning as in Cauchy's Inequality) that the discriminant $\left({2 B}\right)^2 - 4 A C$ of this polynomial must be non-positive.

Thus:
 * $B^2 \le A C$

and hence the result

Also known as
This theorem is also known as the Cauchy-Schwarz Inequality.

It was first stated in this form by Bunyakovsky in 1859, and later rediscovered by Schwarz in 1888.