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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 ...
The importance of such functions lies in the fact that their function space is similar to L p spaces, but its members are not required to satisfy any growth restriction on their behavior at the boundary of their domain (at infinity if the domain is unbounded): in other words, locally integrable functions can grow arbitrarily fast at the domain ...
Conversely, if f is in L p (μ) and g is in L q (μ), then the pointwise product fg is in L 1 (μ). Hölder's inequality is used to prove the Minkowski inequality, which is the triangle inequality in the space L p (μ), and also to establish that L q (μ) is the dual space of L p (μ) for p ∈ [1, ∞).
In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (strong derivative) for functions not assumed differentiable, but only integrable, i.e., to lie in the L p space ([,]).
In mathematics, a uniformly smooth space is a normed ... the L p-spaces are ... it follows that the class of Banach spaces that admit an equivalent uniformly convex ...
By abuse of notation, it is typical to identify ℓ q with the dual of ℓ p: (ℓ p) * = ℓ q. Then reflexivity is understood by the sequence of identifications (ℓ p) ** = (ℓ q) * = ℓ p. The space c 0 is defined as the space of all sequences converging to zero, with norm identical to ||x|| ∞. It is a closed subspace of ℓ ∞, hence ...
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This page should include examples of functions which are contained in one L^p space, but not another. This math.stackexchange post has a useful discussion of this question — Preceding unsigned comment added by 63.253.110.78 ( talk ) 18:11, 26 July 2018 (UTC) [ reply ]