Pointwise convergence

In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function.

It is weaker than uniform convergence, to which it is often compared.

is a topological space, such as the real or complex numbers or a metric space, for example.

is said to converge pointwise to a given function

is said to be the pointwise limit function of the

The definition easily generalizes from sequences to nets

is the unique accumulation point of the net

Sometimes, authors use the term bounded pointwise convergence when there is a constant

[3] This concept is often contrasted with uniform convergence.

That is a stronger statement than the assertion of pointwise convergence: every uniformly convergent sequence is pointwise convergent, to the same limiting function, but some pointwise convergent sequences are not uniformly convergent.

is a sequence of functions defined by

The pointwise limit of a sequence of continuous functions may be a discontinuous function, but only if the convergence is not uniform.

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.

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 nets do and do not converge.

The definition of pointwise convergence meets these conditions and so it induces a topology, called the topology of pointwise convergence, on the set

The topology of pointwise convergence is the same as convergence in the product topology on the space

then the topology of pointwise convergence on

is equal to the subspace topology that it inherits from the product space

is identified as a subset of this Cartesian product via the canonical inclusion map

is compact, then by Tychonoff's theorem, the space

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: let

has a sub-subsequence which itself converges almost everywhere to zero, for example, the subsequence of functions which do not vanish at

But at no point does the original sequence converge pointwise to zero.

Hence, unlike convergence in measure and