Norm Induced by the Inner Product

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

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

It is straightforward to show that properties 1 and 3 of a norm hold (see §5.5.2). 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 ${\bf C}^N$.