Cauchy-Schwarz inequality

Cauchy-Schwarz Inequality. In an inner product space, $$\left\vert{\left \langle {x, y} \right \rangle}\right\vert \le \left\Vert{x}\right\Vert \times \left\Vert{y}\right\Vert$$

Proof. Use the theorem: $$<X,Y> = ‖X‖∙‖Y‖\cos α.$$ Then $$|<X,Y>| = | ‖X‖∙‖Y‖\cos α | $$ $$ = ‖X‖∙‖Y‖|\cos α|,$$ since $|\cos α| < 1$. Hence $$ < ‖X‖∙‖Y‖. $$ $\square$

In the Euclidean space:

In the space of integrable functions: $$ \left({\int_a^b f (t) g(t) \mathrm d t}\right)^2 \le \int_a^b f(t)^2 \mathrm d t \int_a^b g (t)^2 \mathrm d t$$