POINTWISE CONVERGENCE Meaning and
Definition
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Pointwise convergence is a concept used in mathematics to describe the behavior of a sequence of functions. It refers to the idea that, as the index of the sequence increases, the values of the functions at each point in the domain of definition approach a certain limit, potentially different for every point.
More precisely, given a sequence of functions (f_n) defined on a set E, the sequence is said to converge pointwise to a function f on E if, for every point x in E, the limit of f_n(x) as n approaches infinity is equal to f(x). In other words, for each fixed x, if we consider the values of f_n(x) for increasingly large values of n, these values will eventually approach f(x).
This concept is defined for each individual point separately, without any requirement that the convergence occurs uniformly or simultaneously across the entire space. Hence, the behavior of the sequence at one point does not necessarily depend on the behavior at other points.
Pointwise convergence is a fundamental concept in analysis, providing a basic notion of convergence for functions. It allows for the study of functions and sequences in a local manner, whereby the behavior of the sequence at each point can be examined independently. It is particularly useful when dealing with sequences of functions that do not converge uniformly, as pointwise convergence offers a weaker notion of convergence that is still valuable for many applications.
Etymology of POINTWISE CONVERGENCE
The word "pointwise convergence" is derived from the two terms "pointwise" and "convergence".
The term "pointwise" refers to a property that is evaluated separately at each point in a given set. In mathematics, this term is commonly used to describe a type of convergence where a sequence of functions converges at each individual point in the domain.
The term "convergence" refers to the idea of a sequence or a series approaching a particular limit or value. In the context of pointwise convergence, it means that a sequence of functions converges to a specific value at every point in the domain.
Therefore, the etymology of the term "pointwise convergence" comes from combining these two concepts to describe the notion of a sequence of functions approaching a limit at each point individually.
