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

 * $\displaystyle \forall x: 0 \le \left({x f \left({t}\right) + g \left({t}\right)}\right)^2$.

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


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


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

As the Quadratic Equation $A x^2 + 2 B x + C$ is positive for all $x$, it follows that (using the same reasoning as in Cauchy's Inequality) $B^2 \le 4 A C$.

Hence the result.

Alternative names
This theorem is also known as the Cauchy-Bunyakovsky-Schwarz Inequality.

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