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


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


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

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 $(2B)^2 - 4AC$ of this polynomial must be non-positive, and so $B^2 \le A C$.

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.