Pointwise convergence

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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.[1][2]

Definition[edit]

Suppose that is a set and is a topological space, such as the real or complex numbers or a metric space, for example. A net or sequence of functions all having the same domain and codomain is said to converge pointwise to a given function often written as

if (and only if)
The function is said to be the pointwise limit function of the

Properties[edit]

This concept is often contrasted with uniform convergence. To say that

means that
where is the common domain of and 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. For example, if is a sequence of functions defined by then pointwise on the interval but not uniformly.

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

takes the value when is an integer and when is not an integer, and so is discontinuous at every integer.

The values of the functions 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[edit]

Let denote the set of all functions from some given set 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 what 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 of all functions of the form A net in converges in this topology if an only if it converges points.

The topology of pointwise convergence is the same as convergence in the product topology on the space where is the domain and is the codomain. Explicitly, if is a set of functions from some set into some topological space then the topology of pointwise convergence on is equal to the subspace topology that it inherits from the product space when is identified as a subset if this Cartesian product via the canonical inclusion map defined by

If the codomain is compact, then by Tychonoff's theorem, the space is also compact.

Almost everywhere convergence[edit]

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, which are defined using the floor function: let and mod and let

Then any subsequence of the sequence 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 convergence, pointwise convergence almost everywhere is not the convergence of any topology on the space of functions.

See also[edit]

References[edit]

  1. ^ Rudin, Walter (1976). Principles of Mathematical Analysis. McGraw-Hill. ISBN 0-07-054235-X.
  2. ^ Munkres, James R. (2000). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.