Pointwise convergence
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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 if that makes anyone feel good). Consider the statement
- <math>\lim_{n\rightarrow\infty}f_n(x)=f(x).<math>
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
- <math>\lim_{n\rightarrow\infty}f_n(x)=f(x)\ \mbox{pointwise}.<math>
This concept is often contrasted with uniform convergence. To say that
- <math>\lim_{n\rightarrow\infty}f_n(x)=f(x)\ \mbox{uniformly}<math>
means that
- <math>\lim_{n\rightarrow\infty}\,\sup\{\,\left|f_n(x)-f(x)\right|: x\in\mbox{the domain}\,\}=0.<math>
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
- <math>\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).<math>
The pointwise limit of a sequence of continuous functions may be a discontinuous function, but only if the convergence is not uniform.
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.
Pointwise convergence may also be formulated as convergence in the topology which arises from the seminorm given by
- <math>||f||_x=|f(x)|<math>
the space of functions with this topology is called the space of pointwise convergence.de:Punktweise Konvergenz