In mathematics, pointwise convergence is one of various senses in which a sequence of functionss can converge to a particular function. It is weaker than uniform convergence, to which it is often compared.

Definition

is a topological space, such as the real or complex numbers or a metric space, for example. A sequence of functions and codomain is said to converge pointwise if (and only if) the limit of the sequence is said to be the pointwise limit

The definition easily generalizes from sequences to nets

Sometimes, authors use the term bounded pointwise convergence .

Properties

This concept is often contrasted with uniform convergence stands for the supremum. but not uniformly.

The pointwise limit of a sequence of continuous functions is an integer is not an integer, and so is discontinuous at every integer.

need not be real numbers, but may be in any topological space, in order that the concept of pointwise convergence make sense. Uniform convergence, on the other hand, does not make sense for functions taking values in topological spaces generally, but makes sense for functions taking values in metric spaces, and, more generally, in uniform spaces.

Topology

into some topological space As described in the article on characterizations of the category of topological spaces, if certain conditions are met then it is possible to define a unique topology on a set in terms of which netss do and do not converge. The definition of pointwise convergence meets these conditions and so it induces a topology, called the converges in this topology if and only if it converges pointwise.

The topology of pointwise convergence is the same as convergence in the product topology is equal to the subspace topology that it inherits from the product space

is compact, then by Tychonoff's theorem is also compact.

Almost everywhere convergence

In measure theory, one talks about almost everywhere convergence of a sequence of measurable functions defined on a measurable space. That means pointwise convergence almost everywhere, that is, on a subset of the domain whose complement has measure zero. Egorov's theorem states that pointwise convergence almost everywhere on a set of finite measure implies uniform convergence on a slightly smaller set.

Almost everywhere pointwise convergence on the space of functions on a measure space does not define the structure of a topology on the space of measurable functions on a measure space (although it is a convergence structure). For in a topological space, when every subsequence of a sequence has itself a subsequence with the same subsequential limit, the sequence itself must converge to that limit.

But consider the sequence of so-called "galloping rectangles" functions (also known as the typewriter sequence), which are defined using the floor function mod

But at no point does the original sequence converge pointwise to zero. Hence, unlike convergence in measure and L^p convergence, pointwise convergence almost everywhere is not the convergence of any topology on the space of functions.

See also

References

hu:Függvénysorozatok konvergenciája#Pontonkénti konvergencia