Cauchy-Schwarz Inequality

Semi-inner Product Spaces

Let $V$ be a semi-inner product space over $\mathbb K$ where $\mathbb K$ is a subfield of $\C$.

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

Then:

$\left\vert{\left \langle {x, y} \right \rangle}\right\vert^2 \le \left \langle {x, x} \right \rangle \left \langle {y, y} \right \rangle$

Cauchy's Inequality

The special case of the Cauchy-Schwarz Inequality in a Euclidean space is called Cauchy's Inequality. It was Cauchy who first published this result in 1821.

It is usually stated as:

$\displaystyle \sum {r_i^2} \sum {s_i^2} \ge \left({\sum {r_i s_i}}\right)^2$

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

Complex Numbers

$\displaystyle \left({\sum \left\vert{w_i}\right\vert^2}\right) \left({\sum \left\vert{z_i}\right\vert^2}\right) \ge \left\vert{\sum w_i z_i}\right\vert^2$

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 $\left[{a \, . \, . \, b}\right]$.

Then:

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

Alternative names

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

Source of Name

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

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