Norm Properties

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

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

The first property, ``positivity,'' says only the zero vector has norm zero. The second property is ``subadditivity'' and is sometimes called the ``triangle inequality'' for reasons which can be seen by studying Fig. 5.3. The third property says the norm is ``absolutely homogeneous'' with respect to scalar multiplication (which can be complex, in which case the phase of the scalar has no effect).