Punktoperator

Punktoperator, or point operator, is a term used in functional analysis to denote the evaluation-at-a-point functional on a space of functions. For a fixed point x0 in the domain, the Punktoperator E_x0 assigns to each function f its value at x0, that is E_x0(f) = f(x0). It is a linear functional, mapping from a function space F into the underlying field (typically the real or complex numbers).

The continuity of the Punktoperator depends on the function space. On spaces of continuous functions with the uniform (sup) norm, such as C(K) for a compact K, the evaluation functional is bounded with operator norm 1: |f(x0)| ≤ ||f||∞. In contrast, on spaces where functions are only defined almost everywhere, such as L^p spaces with p < ∞ or L^∞, point evaluation is not a well-defined continuous linear functional, since two representatives of a function that agree a.e. can differ at a point.

Variants include evaluation at a finite set. Given points x1, ..., xm, the vector-valued Punktoperator maps f to (f(x1), ..., f(xm)) in R^m or C^m and is bounded with norm at most 1 under the supremum norm when defined on spaces where point values are meaningful.

Related concepts include the evaluation functional in reproducing kernel Hilbert spaces, where f(x) is represented as an inner product with a reproducing kernel k(·, x). The Dirac delta distribution generalizes point evaluation to spaces of distributions rather than ordinary functions.

Applications of the Punktoperator appear in interpolation, approximation theory, and the study of function spaces, where pointwise values play a critical role.