Editing Lp space (section)

==Applications==

===Statistics===

In statistics, measures of [[central tendency]] and [[statistical dispersion]], such as the [[mean]], [[median]], and [[standard deviation]], can be defined in terms of <math>L^p</math> metrics, and measures of central tendency can be characterized as [[Central tendency#Solutions to variational problems|solutions to variational problems]].

In [[penalized regression]], "L1 penalty" and "L2 penalty" refer to penalizing either the [[Taxicab geometry|<math>L^1</math> norm]] of a solution's vector of parameter values (i.e. the sum of its absolute values), or its squared <math>L^2</math> norm (its [[Euclidean norm|Euclidean length]]). Techniques which use an L1 penalty, like [[LASSO]], encourage sparse solutions (where the many parameters are zero).<ref>{{cite book |last=Hastie |first=T. J. |authorlink=Trevor Hastie |last2=Tibshirani |first2=R. |author2link=Robert Tibshirani |last3=Wainwright |first3=M. J. |year=2015 |title=Statistical Learning with Sparsity: The Lasso and Generalizations |location= |publisher=CRC Press |isbn=978-1-4987-1216-3 }}</ref> [[Elastic net regularization]] uses a penalty term that is a combination of the <math>L^1</math> norm and the squared <math>L^2</math> norm of the parameter vector.

===Hausdorff–Young inequality===

The [[Fourier transform]] for the real line (or, for [[periodic functions]], see [[Fourier series]]), maps <math>L^p(\Reals)</math> to <math>L^q(\Reals)</math> (or <math>L^p(\mathbf{T})</math> to <math>\ell^q</math>) respectively, where <math>1 \leq p \leq 2</math> and <math>\tfrac{1}{p} + \tfrac{1}{q} = 1.</math>  This is a consequence of the [[Riesz–Thorin_theorem#Hausdorff–Young_inequality|Riesz–Thorin interpolation theorem]], and is made precise with the [[Hausdorff–Young inequality]].

By contrast, if <math>p > 2,</math> the Fourier transform does not map into <math>L^q.</math>

===Hilbert spaces===
[[Hilbert space]]s are central to many applications, from [[quantum mechanics]] to [[stochastic calculus]]. The spaces <math>L^2</math> and <math>\ell^2</math> are both Hilbert spaces. In fact, by choosing a Hilbert basis <math>E,</math> i.e., a maximal orthonormal subset of <math>L^2</math> or any Hilbert space, one sees that every Hilbert space is isometrically isomorphic to <math>\ell^2(E)</math> (same <math>E</math> as above), i.e., a Hilbert space of type <math>\ell^2.</math>