Funktionsraum

Funktionsraum, literally “function space” in German, refers to a mathematical object consisting of functions from a set X to a field F (usually R or C) that forms a vector space under pointwise addition and scalar multiplication and is often endowed with additional structure such as a norm, inner product, or topology.

Common examples include C(K) (continuous functions on a compact set K), C([a,b]), L^p spaces on measure spaces, the space of bounded functions B(X), and Sobolev spaces W^{k,p}(Ω). These spaces are used to study properties of functions, such as continuity, integrability, differentiability, and growth.

Topological and geometric properties are central: if a function space is equipped with a norm and is complete, it is a Banach space; if it also carries an inner product inducing the norm, it is a Hilbert space (as in L^2). The choice of norm or inner product determines notions of convergence and stability for sequences and series of functions.

Dual spaces are an important aspect: the continuous dual of L^p is L^q with 1 ≤ p < ∞ and 1/p + 1/q = 1; the dual of C(K) is represented by regular Borel measures on K (via the Riesz representation theorem). These dualities underpin many approximation and optimization techniques.

Applications of Funktionsräume include the analysis of linear operators, approximation theory, Fourier and harmonic analysis, and the study of partial differential equations. In German-language mathematics, Funktionsraum serves as a general term for a function space, with specific spaces named similarly to their English counterparts.