Norm Induced by the Inner Product

We may define a norm on using the inner product:

It is straightforward to show that properties 1 and 3 of a norm hold (see §5.8.2). Property 2 follows easily from the Schwarz Inequality which is derived in the following subsection. Alternatively, we can simply observe that the inner product induces the well known norm on .

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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.