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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 ...
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 ...
Remark: The pointwise convergence and uniform boundedness of the sequence can be relaxed to hold only μ-almost everywhere, provided the measure space (S, Σ, μ) is complete or f is chosen as a measurable function which agrees μ-almost everywhere with the μ-almost everywhere existing pointwise limit.
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, , the (real or complex) vector space of bounded sequences with the supremum norm, and = (,,), the vector space of essentially bounded measurable functions with the essential supremum norm, are two closely related Banach spaces. In fact the former is a special case of the latter.
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The vector space of (equivalence classes of) square integrable functions (with respect to Lebesgue measure) forms the space with = Among the L p {\displaystyle L^{p}} spaces, the class of square integrable functions is unique in being compatible with an inner product , which allows notions like angle and orthogonality to be defined.