LebesgueRäume

Lebesgue-Räume, commonly called L^p-Räume, are a fundamental family of function spaces in measure theory and functional analysis. They are built on a measure space (X, Σ, μ), often with μ being the Lebesgue measure on R^n. For 1 ≤ p < ∞, the L^p(μ) space consists of all μ-measurable functions f: X → R (or C) for which the p-th power is integrable, that is ∫_X |f|^p dμ < ∞. The p-norm is defined by ||f||_p = (∫_X |f|^p dμ)^{1/p}. For p = ∞, L^∞(μ) contains those essentially bounded functions, with norm ||f||_∞ = ess sup_{x∈X} |f(x)|.

L^p spaces are complete, making them Banach spaces. In the special case p = 2, L^2(μ) is a Hilbert space with the inner product ⟨f, g⟩ = ∫_X f g dμ. Key inequalities such as Hölder’s inequality and Minkowski’s inequality underpin their analytic use. The dual space of L^p for 1 < p < ∞ is L^q, where 1/p + 1/q = 1; in particular, (L^1)* ≅ L^∞, while L^∞* properly contains L^1.

Relationships between L^p spaces depend on the measure μ. If μ(X) is finite, L^q ⊆ L^p for q > p, but this inclusion may fail on spaces with infinite measure. L^p spaces provide a natural setting for problems in analysis, probability (as spaces of random variables with finite p-th moment), partial differential equations, and signal processing. Common examples include L^p(R^n) with Lebesgue measure and L^p[0,1] on finite intervals. The concept, named after Henri Lebesgue, underpins much of modern analysis.