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 $$\left|{\left \langle {x, y} \right \rangle}\right|^2 \leq \left\|{x}\right\| \times \left\|{y}\right\|$$.

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.