Editing Lp space (section)

===The {{math|''p''}}-norm in finite dimensions===
[[Image:Vector-p-Norms qtl1.svg|thumb|right|Illustrations of [[unit circle]]s (see also [[superellipse]]) in <math>\Reals^2</math> based on different <math>p</math>-norms (every vector from the origin to the unit circle has a length of one, the length being calculated with length-formula of the corresponding <math>p</math>).]]

The Euclidean length of a vector <math>x = (x_1, x_2, \dots, x_n)</math> in the <math>n</math>-dimensional [[real number|real]] [[vector space]] <math>\Reals^n</math> is given by the [[Euclidean norm]]:
<math display="block">\|x\|_2 = \left({x_1}^2 + {x_2}^2 + \dotsb + {x_n}^2\right)^{1/2}.</math>

The Euclidean distance between two points <math>x</math> and <math>y</math> is the length <math>\|x - y\|_2</math> 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 [[taxicab geometry|rectilinear distance]], which takes into account that streets are either orthogonal or parallel to each other. The class of <math>p</math>-norms generalizes these two examples and has an abundance of applications in many parts of [[mathematics]], [[physics]], and [[computer science]].

For a [[real number]] <math>p \geq 1,</math> the '''<math>p</math>-norm''' or '''<math>L^p</math>-norm''' of <math>x</math> is defined by
<math display="block">\|x\|_p = \left(|x_1|^p + |x_2|^p + \dotsb + |x_n|^p\right)^{1/p}.</math>
The absolute value bars can be dropped when <math>p</math> is a rational number with an even numerator in its reduced form, and <math>x</math> 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 <math>2</math>-norm, and the <math>1</math>-norm is the norm that corresponds to the [[taxicab geometry|rectilinear distance]].

The '''<math>L^\infty</math>-norm''' or [[Chebyshev distance|maximum norm]] (or uniform norm) is the limit of the <math>L^p</math>-norms for <math>p \to \infty</math>, given by:
<math display="block">\|x\|_\infty = \max \left\{|x_1|, |x_2|, \dotsc, |x_n|\right\}</math>

For all <math>p \geq 1,</math> the <math>p</math>-norms and maximum norm satisfy the properties of a "length function" (or [[norm (mathematics)|norm]]), that is:
*only the zero vector has zero length,
*the length of the vector is positive homogeneous with respect to multiplication by a scalar ([[Euler's homogeneous function theorem|positive homogeneity]]), and
*the length of the sum of two vectors is no larger than the sum of lengths of the vectors ([[triangle inequality]]).
Abstractly speaking, this means that <math>\Reals^n</math> together with the <math>p</math>-norm is a [[normed vector space]]. Moreover, it turns out that this space is [[Complete_metric_space|complete]], thus making it a [[Banach space]].

====Relations between {{math|''p''}}-norms====

The grid distance or rectilinear distance (sometimes called the "[[Manhattan distance]]") between two points is never shorter than the length of the line segment between them (the Euclidean or "as the crow flies" distance). Formally, this means that the Euclidean norm of any vector is bounded by its 1-norm:
<math display="block">\|x\|_2 \leq \|x\|_1 .</math>

This fact generalizes to <math>p</math>-norms in that the <math>p</math>-norm <math>\|x\|_p</math> of any given vector <math>x</math> does not grow with <math>p</math>:
{{block indent | em = 1.5 | text = <math>\|x\|_{p+a} \leq \|x\|_p</math> for any vector <math>x</math> and real numbers <math>p \geq 1</math> and <math>a \geq 0.</math> (In fact this remains true for <math>0 < p < 1</math> and <math>a \geq 0</math> .)}}

For the opposite direction, the following relation between the <math>1</math>-norm and the <math>2</math>-norm is known:
<math display="block">\|x\|_1 \leq \sqrt{n} \|x\|_2 ~.</math>

This inequality depends on the dimension <math>n</math> of the underlying vector space and follows directly from the [[Cauchy–Schwarz inequality]].

In general, for vectors in <math>\Complex^n</math> where <math>0 < r < p:</math>
<math display="block">\|x\|_p \leq \|x\|_r \leq n^{\frac{1}{r} - \frac{1}{p}} \|x\|_p ~.</math>

This is a consequence of [[Hölder's inequality]].

====When {{math|0 < ''p'' < 1}}====

[[Image:Astroid.svg|thumb|right|[[Astroid]], unit circle in <math>p = \tfrac{2}{3}</math> metric]]
In <math>\Reals^n</math> for <math>n > 1,</math> the formula
<math display="block">\|x\|_p = \left(|x_1|^p + |x_2| ^p + \cdots + |x_n|^p\right)^{1/p}</math>
defines an absolutely [[homogeneous function]] for <math>0 < p < 1;</math> however, the resulting function does not define a norm, because it is not [[subadditivity|subadditive]]. On the other hand, the formula
<math display="block">|x_1|^p + |x_2|^p + \dotsb + |x_n|^p</math>
defines a subadditive function at the cost of losing absolute homogeneity. It does define an [[F-space|F-norm]], though, which is homogeneous of degree <math>p.</math>

Hence, the function
<math display="block">d_p(x, y) = \sum_{i=1}^n |x_i - y_i|^p</math>
defines a [[metric space|metric]]. The [[metric space]] <math>(\Reals^n, d_p)</math> is denoted by <math>\ell_n^p.</math>

Although the <math>p</math>-unit ball <math>B_n^p</math> around the origin in this metric is "concave", the topology defined on <math>\Reals^n</math> by the metric <math>B_p</math> is the usual vector space topology of <math>\Reals^n,</math> hence <math>\ell_n^p</math> is a [[locally convex]] topological vector space. Beyond this qualitative statement, a quantitative way to measure the lack of convexity of <math>\ell_n^p</math> is to denote by <math>C_p(n)</math> the smallest constant <math>C</math> such that the scalar multiple <math>C \, B_n^p</math> of the <math>p</math>-unit ball contains the convex hull of <math>B_n^p,</math> which is equal to <math>B_n^1.</math> The fact that for fixed <math>p < 1</math> we have
<math display="block">C_p(n) = n^{\tfrac{1}{p} - 1} \to \infty, \quad \text{as } n \to \infty</math>
shows that the infinite-dimensional sequence space <math>\ell^p</math> defined below, is no longer locally convex.{{citation needed|date=November 2015}}

====When {{math|1=''p'' = 0}}====

There is one <math>\ell_0</math> norm and another function called the <math>\ell_0</math> "norm" (with quotation marks).

The mathematical definition of the <math>\ell_0</math> norm was established by [[Stefan Banach|Banach]]'s ''[[Theory of Linear Operations]]''. The [[F-space|space]] of sequences has a complete metric topology provided by the [[F-space|F-norm]] on the [[Metric_space#Product_metric_spaces|product metric]]:{{Citation needed|date=December 2024}}
<math display="block">(x_n) \mapsto \|x\|:=d(0,x)=\sum_n 2^{-n} \frac{|x_n|}{1 +|x_n|}.</math> 
The <math>\ell_0</math>-normed space is studied in functional analysis, probability theory, and harmonic analysis.

Another function was called the <math>\ell_0</math> "norm" by [[David Donoho]]—whose quotation marks warn that this function is not a proper norm—is the number of non-zero entries of the vector <math>x.</math>{{Citation needed|date=September 2022}} Many authors [[abuse of terminology|abuse terminology]] by omitting the quotation marks. Defining [[zero to the power of zero|<math>0^0 = 0,</math>]] the zero "norm" of <math>x</math> is equal to
<math display="block">|x_1|^0 + |x_2|^0 + \cdots + |x_n|^0 .</math>

[[File:Lp space animation.gif|alt=An animated gif of unit circles in p-norms 0.1 through 2 with a step of 0.05.|thumb|An animated gif of p-norms 0.1 through 2 with a step of 0.05.]]
This is not a [[norm (mathematics)|norm]] because it is not [[Homogeneous function|homogeneous]]. For example, scaling the vector <math>x</math> by a positive constant does not change the "norm". Despite these defects as a mathematical norm, the non-zero counting "norm" has uses in [[scientific computing]], [[information theory]], and [[statistics]]–notably in [[compressed sensing]] in [[signal processing]] and computational [[harmonic analysis]]. Despite not being a norm, the associated metric, known as [[Hamming distance]], is a valid distance, since homogeneity is not required for distances.