BACKGROUND MATERIAL

For a vector $x=(x_1, \cdots, x_n)^t \in \Re^n$, we define the norm and inner product

$$\Vert x\Vert := (x_1^2+ \cdots+ x_n^2)^{1/2},$$

$$x \cdot y := x_1y_1+ \cdots+ x_ny_n.$$

Theorem 1 (Cauchy-Schwarz)   $\vert x \cdot y\vert \leq \Vert x\Vert\Vert y\Vert,$ with equality if and only if $x=\lambda y,$ for some $\lambda.$