Lp

Lp refers to the family of Lebesgue spaces, function spaces defined on a measure space. For a measure space (X, Σ, μ) and a real number p with 1 ≤ p < ∞, Lp(X, μ) consists of all measurable functions f: X → ℂ (or ℝ) for which the p-th power is integrable, meaning ∫X |f(x)|^p dμ(x) < ∞. Functions that differ only on a set of measure zero are identified in Lp, so elements are equivalence classes under almost everywhere equality. The space is equipped with the norm ||f||p = (∫X |f|^p dμ)1/p. For p = ∞, L∞(X, μ) is the space of essentially bounded functions with the norm ||f||∞ = ess supX |f(x)|.

Lp spaces are Banach spaces: complete normed spaces, and they play a central role in analysis. For 1 < p < ∞, the dual space of Lp is Lq, where 1/p + 1/q = 1. In particular, L2 is a Hilbert space with inner product ⟨f, g⟩ = ∫X f(x) overline{g(x)} dμ(x). L1 consists of integrable functions, while L∞ consists of essentially bounded functions.

Common embeddings depend on the measure of X. If μ(X) is finite, then Lq ⊂ Lp for q ≥ p, with ||f||p ≤ μ(X)1/p−1/q ||f||q. In spaces with infinite measure, inclusions can reverse. Variants include weighted Lp spaces and Lp spaces on other measure spaces, such as Lebesgue measure on ℝn. Lp spaces are used across analysis, probability theory, Fourier analysis, and partial differential equations due to their structural and approximation properties.