L2Funktionsräumen

L2Funktionsräumen, also known as L2 spaces, are a fundamental concept in functional analysis, a branch of mathematics that studies spaces of functions. These spaces are defined over a measure space (Ω, Σ, μ), where Ω is the sample space, Σ is a σ-algebra of subsets of Ω, and μ is a measure on Σ. The L2 space, denoted as L2(Ω, Σ, μ), consists of all measurable functions f: Ω → ℂ (the complex numbers) such that the integral of the absolute square of f is finite. This is expressed mathematically as:

∫|f|^2 dμ < ∞

The L2 space is equipped with an inner product, defined by:

⟨f, g⟩ = ∫f * g dμ

where f* denotes the complex conjugate of f. This inner product induces a norm, given by:

||f|| = ∫|f|^2 dμ

The L2 space is a Hilbert space, meaning it is complete with respect to this norm and the inner product satisfies the Cauchy-Schwarz inequality. L2 spaces are widely used in various areas of mathematics and physics, including Fourier analysis, quantum mechanics, and signal processing. They provide a natural setting for studying functions and their properties, such as orthogonality and completeness.