Cauchy-Bunyakovsky-Schwarz Inequality/Inner Product Spaces

Theorem
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 $V$.

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

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