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 in common (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). Consider the statement

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

To say that this is true of each value of x in the domain, separately, is to say that the sequence { fn } converges pointwise to f, and often one writes

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

[edit] Properties

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

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

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 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 may also be formulated as convergence in the topology which arises from the seminorm given by

\|f\|_x=|f(x)|.\,

The space of functions with this topology is called the space of pointwise convergence. This topology is equivalent to the product topology on YX. If the domain and image of f are compact, then, by Tychonoff's theorem, the resulting space 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.