Norm Induced by the Inner Product

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

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

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 $\ensuremath{L_2}$ norm on $\mathbb{C}^N$ .