L2spaces

An L2 space, denoted L^2(X, μ), is the set of square-integrable functions on a measure space (X, Σ, μ). A function f is square-integrable if the integral of its absolute value squared, ∫|f|^2 dμ, is finite. Elements of L^2 are equivalence classes of measurable functions, with two functions identified if they are equal almost everywhere.

L^2 is equipped with the inner product ⟨f, g⟩ = ∫ f(x) overline{g(x)} dμ, and the associated norm is ‖f‖_2 = sqrt(⟨f, f⟩). With this inner product, L^2 becomes a complex Hilbert space: it is complete in the norm induced by the inner product, and standard Hilbert-space tools such as orthogonality, projections, and expansions apply.

Common instances include L^2([0, 1]) and L^2(ℝ) with the Lebesgue measure. L^2 spaces are separable for these standard measures, and they form the natural setting for Fourier analysis and quantum mechanics. Parseval’s identity and the Plancherel theorem relate L^2 norms to Fourier transforms, allowing energy preservation statements in frequency space.

Applications of L^2 spaces span signal processing, statistics, and probability (where square-integrable random variables are studied), as well as the formulation of quantum states in physics. They also underpin many techniques for decomposing functions into orthogonal bases, such as Fourier bases or wavelets, through the structure of orthonormal bases and projections in a Hilbert space.