Cauchy-Bunyakovsky-Schwarz Inequality/Inner Product Spaces

Theorem
Let $V$ be an inner-product space over $\mathbb K$ where $\mathbb K = \R$ or $\C$.

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

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

Proof
Let $\lambda \in \mathbb K$. Since an inner-product is generated by a norm on the underlying normed linear space we may expand as follows:

where $\lambda^*$ is the complex conjugate of $\lambda$.

(If $\mathbb K = \R$, then $\lambda^* = \lambda$.)

If we let $\lambda = \left \langle {x, y} \right \rangle \times \left \langle {y, y} \right \rangle^{-1}$ then we obtain:

$0 \le \left \langle {x, x} \right \rangle - \left|{\left \langle {x, y} \right \rangle}\right|^2 \times \left \langle {y, y} \right \rangle^{-1} $

Solving this for $\left|{\left \langle {x, y} \right \rangle}\right|^2 $, we see that

$\left|{\left \langle {x, y} \right \rangle}\right|^2 \le \left \langle {x, x} \right \rangle * \left \langle {y, y} \right \rangle = \left\|{x}\right\| \times \left\|{y}\right\|$

as desired.

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