Vector Norm

Given an $n$-D Vector

$${\bf x} = \left[{\matrix{x_1\cr x_2\cr \vdots\cr x_n\cr}}\right],$$

a vector norm $\vert\vert{\bf x}\vert\vert$ (sometimes written simply $\vert{\bf x}\vert$) is a Nonnegative number satisfying

1. $\vert\vert x\vert\vert>0$ when ${\bf x}\not={\bf0}$ and $\vert\vert{\bf x}\vert\vert=0$ Iff ${\bf x}={\bf0}$,

2. $\vert\vert k{\bf x}\vert\vert=\vert k\vert\,\vert\vert{\bf x}\vert\vert$ for any Scalar $k$,

3. $\vert\vert{\bf x}+{\bf y}\vert\vert\leq \vert\vert{\bf x}\vert\vert+\vert\vert{\bf y}\vert\vert.$

See also Compatible, Matrix Norm, Natural Norm, Norm

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1114, 1980.