Editing Mercer's theorem (section)

{{Short description|Mathematical theorem}}
{{More footnotes needed|date=December 2024}}
In [[mathematics]], specifically [[functional analysis]], '''Mercer's theorem''' is a representation of a symmetric [[Definite bilinear form|positive-definite]] function on a square as a sum of a convergent sequence of product functions.  This theorem, presented in {{harv|Mercer|1909}}, is one of the most notable results of the work of [[James Mercer (mathematician)|James Mercer]] (1883–1932). It is an important theoretical tool in the theory of [[integral equation]]s; it is used in the [[Hilbert space]] theory of [[stochastic process]]es, for example the [[Karhunen–Loève theorem]]; and it is also used in the [[reproducing kernel Hilbert space]] theory where it characterizes a symmetric [[positive-definite kernel]] as a reproducing kernel.<ref>{{cite web |first=Peter |last=Bartlett |title=Reproducing Kernel Hilbert Spaces |date=2008 |work=Lecture notes of CS281B/Stat241B Statistical Learning Theory |publisher=University of California at Berkeley |url=https://people.eecs.berkeley.edu/~bartlett/courses/281b-sp08/7.pdf }}</ref>