Editing Lp space (section)

====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.