Lp space

In mathematics, the L^p spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue (Dunford & Schwartz 1958, III.3), although according to the Bourbaki group (Bourbaki 1987) they were first introduced by Frigyes Riesz (Riesz 1910).

L^p spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, economics, finance, engineering, and other disciplines.

The Euclidean length of a vector ${\displaystyle x=(x_{1},x_{2},\dots ,x_{n})}$ in the ${\displaystyle n}$-dimensional real vector space ${\displaystyle \mathbb {R} ^{n}}$ is given by the Euclidean norm: $${\displaystyle \|x\|_{2}=\left({x_{1}}^{2}+{x_{2}}^{2}+\dotsb +{x_{n}}^{2}\right)^{1/2}.}$$

The Euclidean distance between two points ${\displaystyle x}$ and ${\displaystyle y}$ is the length ${\displaystyle \|x-y\|_{2}}$ of the straight line between the two points. In many situations, the Euclidean distance is appropriate for capturing the actual distances in a given space. In contrast, consider taxi drivers in a grid street plan who should measure distance not in terms of the length of the straight line to their destination, but in terms of the rectilinear distance, which takes into account that streets are either orthogonal or parallel to each other. The class of ${\displaystyle p}$-norms generalizes these two examples and has an abundance of applications in many parts of mathematics, physics, and computer science.

For a real number ${\displaystyle p\geq 1,}$ the ${\displaystyle p}$-norm or ${\displaystyle L^{p}}$-norm of ${\displaystyle x}$ is defined by $${\displaystyle \|x\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\dotsb +|x_{n}|^{p}\right)^{1/p}.}$$ The absolute value bars can be dropped when ${\displaystyle p}$ is a rational number with an even numerator in its reduced form, and ${\displaystyle x}$ is drawn from the set of real numbers, or one of its subsets.

The Euclidean norm from above falls into this class and is the ${\displaystyle 2}$-norm, and the ${\displaystyle 1}$-norm is the norm that corresponds to the rectilinear distance.

The ${\displaystyle L^{\infty }}$-norm or maximum norm (or uniform norm) is the limit of the ${\displaystyle L^{p}}$-norms for ${\displaystyle p\to \infty }$, given by: $${\displaystyle \|x\|_{\infty }=\max \left\{|x_{1}|,|x_{2}|,\dotsc ,|x_{n}|\right\}}$$

For all ${\displaystyle p\geq 1,}$ the ${\displaystyle p}$-norms and maximum norm satisfy the properties of a "length function" (or norm), that is: