Cauchy-Bunyakovsky-Schwarz Inequality

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Semi-Inner Product Spaces

Let $\mathbb K$ be a subfield of $\C$.

Let $V$ be a semi-inner product space over $\mathbb K$.

Let $x, y$ be vectors in $V$.


$\size {\innerprod x y}^2 \le \innerprod x x \innerprod y y$

Lebesgue $2$-Space

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.


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

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

Complex Numbers

$\ds \paren {\sum \cmod {w_i}^2} \paren {\sum \cmod {z_i}^2} \ge \cmod {\sum w_i z_i}^2$

where all of $w_i, z_i \in \C$.

Definite Integrals

Let $f$ and $g$ be real functions which are continuous on the closed interval $\closedint a b$.


$\ds \paren {\int_a^b \map f t \, \map g t \rd t}^2 \le \int_a^b \paren {\map f t}^2 \rd t \int_a^b \paren {\map g t}^2 \rd t$

Also known as

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.

Also see

The special case of the Cauchy-Bunyakovsky-Schwarz Inequality in a Euclidean space is called Cauchy's Inequality.

It is usually stated as:

$\ds \sum {r_i}^2 \sum {s_i}^2 \ge \paren {\sum {r_i s_i} }^2$

where all of $r_i, s_i \in \R$.

Source of Name

This entry was named for Augustin Louis CauchyKarl Hermann Amandus Schwarz and Viktor Yakovlevich Bunyakovsky.