Norm Properties

There are many other possible choices of norm. To qualify as a norm on , a real-valued signal-function must satisfy the following three properties:

1. , with
2. 
3. ,

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 can be complex, in which case the angle of has no effect).

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.