Cauchy-Schwarz Inequality

Theorem

Let

• $\mathbf{u}=(u_1,u_2,\dots ,u_n)$ vectors in $R^n$
• $\mathbf{v}=(v_1,v_2,\dots ,v_n)$ points vectors in $R^n$

Then

$$|\mathbf{u}\cdot \mathbf{v}|\le ||\mathbf{u}||||\mathbf{v}||$$

Or (in terms of components)

$$u_1{v}_1+u_2{v}_2+\cdots +u_n{v}_n\le (u_1^2+u_2^2+\cdots +u_n^2{)}^{1/2}(v_1^2+v_2^2+\cdots +u_n^2{)}^{1/2}$$