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.

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 ${\displaystyle L^{p}}$ metrics, and measures of central tendency can be characterized as solutions to variational problems.

In penalized regression, "L1 penalty" and "L2 penalty" refer to penalizing either the ${\displaystyle L^{1}}$ norm of a solution's vector of parameter values (i.e. the sum of its absolute values), or its squared ${\displaystyle L^{2}}$ norm (its Euclidean length). Techniques which use an L1 penalty, like LASSO, encourage sparse solutions (where the many parameters are zero).^[1] Elastic net regularization uses a penalty term that is a combination of the ${\displaystyle L^{1}}$ norm and the squared ${\displaystyle L^{2}}$ norm of the parameter vector.

Hausdorff–Young inequality

The Fourier transform for the real line (or, for periodic functions, see Fourier series), maps ${\displaystyle L^{p}(\mathbb {R} )}$ to ${\displaystyle L^{q}(\mathbb {R} )}$ (or ${\displaystyle L^{p}(\mathbf {T} )}$ to ${\displaystyle \ell ^{q}}$) respectively, where ${\displaystyle 1\leq p\leq 2}$ and ${\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1.}$ This is a consequence of the Riesz–Thorin interpolation theorem, and is made precise with the Hausdorff–Young inequality.

By contrast, if ${\displaystyle p>2,}$ the Fourier transform does not map into ${\displaystyle L^{q}.}$

Hilbert spaces

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

The p-norm in finite dimensions

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: $${\displaystyle \Vert <!--\; \Vert \; -->x{\Vert <!--\; \Vert \; -->}_2={({x_1}^2+{x_2}^2+\cdots <!--\; \cdots \; -->+{x_{}}^{}}^{}}$$Zdroj:https://en.wikipedia.org?pojem=Lp_space
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