Normkonvergenz

Normkonvergenz is a concept in functional analysis that describes the convergence of a sequence of functions in a normed vector space. Specifically, a sequence of functions $f_n$ converges in norm to a function $f$ if the norm of the difference between $f_n$ and $f$ approaches zero as $n$ approaches infinity. The norm used is typically a norm defined on the space of functions, such as the supremum norm or the $L^p$ norm.

Norm convergence is a strong form of convergence. If a sequence of functions converges in norm, it also converges pointwise and uniformly (under certain conditions, like the space being a Banach space). However, the converse is not always true. A sequence of functions might converge pointwise or uniformly without converging in norm.

The notion of norm convergence is fundamental in the study of function spaces. For instance, it is used to define completeness of normed spaces, leading to the concept of Banach spaces. Many important theorems in functional analysis, such as the Riesz representation theorem and the closed graph theorem, rely on the properties of norm convergence. It provides a precise way to measure the "distance" between functions and to analyze the behavior of sequences and series of functions. Understanding norm convergence is crucial for working with operators on function spaces and for developing approximation theory.