punktvise

Punktvise, or pointwise, is a term used in analysis to describe a mode of convergence for sequences of functions. It denotes convergence that occurs at each point of the domain independently of the others. The concept is widely used in mathematics to distinguish it from stronger forms of convergence, such as uniform convergence.

Formally, let f_n be a sequence of functions defined on a common domain X with codomain R or C. The sequence f_n converges punktvis to a function f on X if, for every x in X, the sequence of real numbers f_n(x) converges to f(x) as n tends to infinity. This definition imposes no requirement about how fast the convergence happens uniformly across all x in X.

An illustrative example is the sequence f_n(x) = x^n on the interval [0,1]. For each x in [0,1), f_n(x) → 0, and for x = 1, f_n(1) → 1. Thus f_n converges punktvis to the function f, where f(x) = 0 for x in [0,1) and f(1) = 1.

Relation to other convergence notions is central: pointwise convergence is weaker than uniform convergence. Uniform convergence implies that the limit function inherits properties like continuity from the approximating functions, but pointwise convergence alone does not guarantee such preservation. In analysis, pointwise convergence is often contrasted with almost everywhere convergence and convergence in normed spaces.

The term appears in various languages (such as punktwise in German and punktvis in Scandinavian languages) and is commonly abbreviated as “pointwise convergence” in English.