Pointwise convergence

In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function.^[1]^[2]

Definition

Suppose { f_n } is a sequence of functions sharing the same domain and codomain (for the moment, defer specifying the nature of the values of these functions, but the reader may take them to be real numbers). The sequence { f_n } converges pointwise to f, often written as

\lim_{n\rightarrow\infty}f_n=f\ \mbox{pointwise},

if and only if

\lim_{n\rightarrow\infty}f_n(x)=f(x).

for every x in the domain. Also f is said to be the pointwise limit function of {f_n}.

Properties

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

\lim_{n\rightarrow\infty}f_n=f\ \mbox{uniformly}

means that

\lim_{n\rightarrow\infty}\,\sup\{\,\left|f_n(x)-f(x)\right|: x\in\mbox{the domain}\,\}=0.

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

\lim_{n\rightarrow\infty} x^n=0\ \mbox{pointwise}\ \mbox{on}\ \mbox{the}\ \mbox{interval}\ [0,1),\ \mbox{but}\ \mbox{not}\ \mbox{uniformly}\ \mbox{on}\ \mbox{the}\ \mbox{interval}\ [0,1).

as the speed of convergence depends on x and is faster for lower values of x in the domain.

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

f(x)=\lim_{n\rightarrow\infty} \cos (\pi x)^{2n}

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 f_n 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

Pointwise convergence is the same as convergence in the product topology on the space Y^X, where X is the domain and Y is the codomain. If the codomain Y is compact, then, by Tychonoff's theorem, the space Y^X 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. Egorov's theorem states that pointwise convergence almost everywhere on a set of finite measure implies uniform convergence on a slightly smaller set.

References

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

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