Some properties of the scalar product:

• Positive definite: $<\underline{\textbf{x}},\underline{\textbf{x}}> \geq 0$
• Defines the norm: $\vert\vert\underline{\textbf{x}}\vert\vert^2 = <\underline{\textbf{x}},\underline{\textbf{x}}>$
• Hermitic: $<\underline{\textbf{x}},\underline{\textbf{y}}> = (\frac{1}{L}\underline{\textb... ...h \underline{\textbf{x}})^* = <\underline{\textbf{y}},\underline{\textbf{x}}>^*$
• Cauchy-Swartz: $\vert\underline{\textbf{x}}^h \underline{\textbf{y}}\vert^2 \leq \vert\vert\underline{\textbf{x}}\vert\vert^2 \vert\vert\underline{\textbf{y}}\vert\vert^2$