Lp space
Definition

For a real number p ≥ 1, {\displaystyle p\geq 1,} the p {\displaystyle p} -norm or L p {\displaystyle L^{p}} -norm of x {\displaystyle x} is defined by ‖ x ‖ p = (| x 1 | p + | x 2 | p + ⋯ + | x n | p) 1 / p. {\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 p {\displaystyle p} is a rational number with an even numerator in its reduced form, and x {\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 2 {\displaystyle 2} -norm, and the 1 {\displaystyle 1} -norm is the norm that corresponds to the rectilinear distance.

The L ∞ {\displaystyle L^{\infty }} -norm or maximum norm (or uniform norm) is the limit of the L p {\displaystyle L^{p}} -norms for p → ∞. {\displaystyle p\to \infty.} It turns out that this limit is equivalent to the following definition: ‖ x ‖ ∞ = max { | x 1 |, | x 2 |, …, | x n | } {\displaystyle \|x\|_{\infty }=\max \left\{|x_{1}|,|x_{2}|,\dotsc,|x_{n}|\right\}}

See L -infinity.

For all p ≥ 1, {\displaystyle p\geq 1,} the p {\displaystyle p} -norms and maximum norm as defined above indeed satisfy the properties of a "length function" (or norm), which are that:

Abstractly speaking, this means that R n {\displaystyle \mathbb {R} ^{n}} together with the p {\displaystyle p} -norm is a normed vector space. Moreover, it turns out that this space is complete, thus making it a Banach space. This Banach space is the L p {\displaystyle L^{p}} -space over { 1, 2, …, n }. {\displaystyle \{1,2,\ldots,n\}.}
oulipo:avoid_letter[o], combination:two_responses