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Norm Induced by the Inner Product

We may define a norm on $ \underline{u}\in{\bf C}^N$ using the inner product:

$\displaystyle \zbox {\Vert\underline{u}\Vert \isdef \sqrt{\left<\underline{u},\underline{u}\right>}}

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 $ L2$ norm on $ {\bf C}^N$.

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Cauchy-Schwarz Inequality
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Linearity of the Inner Product