funktionalem

Funktional, in mathematics, denotes a linear map from a vector space V over a field F to F itself. The set of all linear functionals on V is called the dual space, denoted V*. If V is finite dimensional, the dual has the same dimension as V and many functionals can be represented by inner products with a fixed vector (via the Riesz representation in inner product spaces).

Examples: For R^n with standard coordinates, the functional f_a(x) = a1 x1 + ... + an x_n is linear and scalar-valued. In spaces of functions, evaluation at a point t, ev_t: f -> f(t), is a linear functional when the function space consists of real- or complex-valued functions. In calculus of variations, functionals map functions to numbers but need not be linear; e.g., the action integral.

Properties: Functionals may be continuous or discontinuous; in normed spaces, continuous linear functionals are bounded. The Hahn–Banach theorem ensures extension of functionals under certain conditions. The Riesz representation theorem provides a representation of continuous linear functionals on Hilbert spaces as inner products with a fixed vector.

Broader usage: In a wider sense, a functional can refer to any map from a space of functions to the real or complex numbers, not necessarily linear; such functionals are central in analysis, for example in the calculus of variations or in integral formulations.

Linguistic note: In German, funktionalem is a declined form of the adjective functional; used in phrases like im funktionalem Sinn or des funktionalen Systems. Thus the term can appear grammatically in forms distinct from the mathematical concept.