Cauchy-Bunyakovsky-Schwarz Inequality/Lebesgue 2-Space

Theorem

Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $f, g: X \to \R$ be $\mu$-square integrable functions, that is $f, g \in \map {\LL^2} \mu$, Lebesgue $2$-space.

Then:

$\ds \int \size {f g} \rd \mu \le \norm f_2^2 \cdot \norm g_2^2$

where $\norm {\, \cdot \,}_2$ is the $2$-norm.

Equality in the above, that is:

$\ds \int \size {f g} \rd \mu \le \norm f_2^2 \cdot \norm g_2^2$

$\dfrac {\size {\map f x}^2} {\norm f_2^2} = \dfrac {\size {\map g x}^2} {\norm g_2^2}$

Follows directly from Hölder's Inequality for Integrals with $p = q = 2$.

$\blacksquare$

The Cauchy-Bunyakovsky-Schwarz Inequality in its various form is also known as:

the Cauchy-Schwarz-Bunyakovsky Inequality

the Cauchy-Schwarz Inequality

Schwarz's Inequality or the Schwarz Inequality

Bunyakovsky's Inequality or Buniakovski's Inequality.

For brevity, it is sometimes referred to by the abbreviations CS inequality or CBS inequality.