# vector p-norm

A class of vector norms, called a $p$-norm and denoted $||\cdot||_{p}$, is defined as

 $||\,x\,||_{p}=(|x_{1}|^{p}+\cdots+|x_{n}|^{p})^{\frac{1}{p}}\qquad p\geq 1,x% \in\mathbbmss{R}^{n}$

The most widely used are the 1-norm, 2-norm, and $\infty$-norm:

 $\displaystyle||\,x\,||_{1}$ $\displaystyle=$ $\displaystyle|x_{1}|+\cdots+|x_{n}|$ $\displaystyle||\,x\,||_{2}$ $\displaystyle=$ $\displaystyle\sqrt{|x_{1}|^{2}+\cdots+|x_{n}|^{2}}=\sqrt{x^{T}x}$ $\displaystyle||\,x\,||_{\infty}$ $\displaystyle=$ $\displaystyle\displaystyle\max_{1\leq i\leq n}|x_{i}|$

The 2-norm is sometimes called the Euclidean vector norm, because $||\,x-y\,||_{2}$ yields the Euclidean distance between any two vectors $x,y\in\mathbbmss{R}^{n}$. The 1-norm is also called the taxicab metric (sometimes Manhattan metric) since the distance of two points can be viewed as the distance a taxi would travel on a city (horizontal and vertical movements).

A useful fact is that for finite dimensional spaces (like $\mathbbmss{R}^{n}$) the three mentioned norms are http://planetmath.org/node/4312equivalent. Moreover, all $p$-norms are equivalent. This can be proved using that any norm has to be continuous in the $2$-norm and working in the unit circle.

The $L^{p}$-norm (http://planetmath.org/LpSpace) in function spaces is a generalization of these norms by using counting measure.

 Title vector p-norm Canonical name VectorPnorm Date of creation 2013-03-22 11:43:03 Last modified on 2013-03-22 11:43:03 Owner Andrea Ambrosio (7332) Last modified by Andrea Ambrosio (7332) Numerical id 21 Author Andrea Ambrosio (7332) Entry type Definition Classification msc 46B20 Classification msc 05Cxx Classification msc 05-01 Classification msc 20H15 Classification msc 20B30 Synonym Minkowski norm Synonym Euclidean vector norm Synonym vector Euclidean norm Synonym vector 1-norm Synonym vector 2-norm Synonym vector infinity-norm Synonym L^p metric Synonym L^p Related topic VectorNorm Related topic CauchySchwartzInequality Related topic HolderInequality Related topic FrobeniusMatrixNorm Related topic LpSpace Related topic CauchySchwarzInequality Defines Manhattan metric Defines Taxicab Defines L^1 norm Defines L^1 metric Defines L^2 metric Defines L^2 norm Defines L^∞norm