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.
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[edit] Definition
Suppose { fn } is a sequence of functions sharing the same domain (for the moment, we defer making precise the nature of the values of these functions, but the reader may take them to be real numbers). The sequence { fn } converges pointwise to f, often written as
if and only if
for every x in the domain.
[edit] Properties
This concept is often contrasted with uniform convergence. To say that
means that
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 we have
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 1 when x is an integer and 0 when x is not an integer, and so is discontinuous at every integer.
The values of the functions fn 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.
[edit] Topology
Pointwise convergence is the same as convergence in the product topology on the space YX. If Y is compact, then, by Tychonoff's theorem, then the space YX is also compact.
[edit] In measure theory
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. Egorov's theorem states that pointwise convergence almost everywhere on a set of finite measure implies uniform convergence on a slightly smaller set.
