Cauchy-Bunyakovsky-Schwarz Inequality

Cauchy-Schwartz Inequality
(AKA: Schwartz inequality or Cauchy–Bunyakovsky–Schwarz inequality)

Let $$V$$ be an inner-product space over $$\mathbb{K}$$ where $$\mathbb{K}=\mathbb{R}$$ or $$\mathbb{C}$$). Let $$x,y$$ be vectors in $$V$$.

Theorem:$$||^2 \leq \| x\| \times \| y\|$$

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

$$ 0 \leq \| x-\lambda y\| =  = --<\lambda y ,x>+<\lambda y,\lambda y> = -\lambda^* -\lambda +<\lambda y , \lambda y> $$

where $$\lambda^*$$ is the complex conjugate of $$\lambda$$ ( if $$\mathbb{K}=\mathbb{R}$$, then $$\lambda^*=\lambda$$).

If we let $$\lambda =  \times ^{-1}$$ then we obtain:

$$ 0 \leq  - ||^2 \times^{-1} $$

Solving this for $$ ||^2 $$, we see that

$$||^2 \leq *= \| x\| \times \| y\|$$

as desired.