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Lp space

In mathematics, the Lp 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). Lp 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, finance, engineering, and other disciplines. In mathematics, the Lp 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). Lp 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, finance, engineering, and other disciplines. In statistics, measures of central tendency and statistical dispersion, such as the mean, median, and standard deviation, are defined in terms of Lp 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 L1 norm of a solution's vector of parameter values (i.e. the sum of its absolute values), or its L2 norm (its Euclidean length). Techniques which use an L1 penalty, like LASSO, encourage solutions where many parameters are zero. Techniques which use an L2 penalty, like ridge regression, encourage solutions where most parameter values are small. Elastic net regularization uses a penalty term that is a combination of the L1 norm and the L2 norm of the parameter vector. The Fourier transform for the real line (or, for periodic functions, see Fourier series), maps Lp(R) to Lq(R) (or Lp(T) to ℓq) respectively, where 1 ≤ p ≤ 2 and 1/p + 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 p > 2, the Fourier transform does not map into Lq. Hilbert spaces are central to many applications, from quantum mechanics to stochastic calculus. The spaces L2 and ℓ2 are both Hilbert spaces. In fact, by choosing a Hilbert basis (i.e., a maximal orthonormal subset of L2 or any Hilbert space), one sees that all Hilbert spaces are isometric to ℓ2(E), where E is a set with an appropriate cardinality. The length of a vector x = (x1, x2, ..., xn) in the n-dimensional real vector space Rn is usually given by the Euclidean norm: The Euclidean distance between two points x and y is the length ||x − y||2 of the straight line between the two points. In many situations, the Euclidean distance is insufficient for capturing the actual distances in a given space. An analogy to this is suggested by 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 rectilinear distance, which takes into account that streets are either orthogonal or parallel to each other. The class of p-norms generalizes these two examples and has an abundance of applications in many parts of mathematics, physics, and computer science. For a real number p ≥ 1, the p-norm or Lp-norm of x is defined by

[ "Functional analysis", "Banach space", "Bochner's theorem", "Palais–Smale compactness condition", "Euclid's lemma", "Discontinuous linear map", "ba space" ]
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