Norm Properties

There are many other possible choices of norm. To qualify as a norm on $\mathbb{C}^N$ , a real-valued signal-function $f(\underline{x})$ must satisfy the following three properties:

1. $f(\underline{x})\ge 0$ , with $0\Leftrightarrow \underline{x}=\underline{0}$
2. $f(\underline{x}+\underline{y})\leq f(\underline{x})+f(\underline{y})$
3. $f(c\underline{x}) = \left\vert c\right\vert f(\underline{x})$ , $\forall c\in\mathbb{C}$

The first property, ``positivity,'' says the norm is nonnegative, and only the zero vector has norm zero. The second property is ``subadditivity'' and is sometimes called the ``triangle inequality'' for reasons that can be seen by studying Fig.5.6. The third property says the norm is ``absolutely homogeneous'' with respect to scalar multiplication. (The scalar $c$ can be complex, in which case the angle of $c$ has no effect).