In [[functional analysis]] (a branch of [[mathematics]]), a '''reproducing kernel [[Hilbert space]]''' is a [[function space]] in which pointwise evaluation is a [[bounded operator|continuous linear functional]]. Equivalently, they are spaces that can be defined by [[reproducing kernel]]s. The subject was originally and simultaneously developed by [[N. Aronszajn]] and [[S. Bergman]] in [[1950]].
In this article we assume that [[Hilbert spacesspace]]s are [[complex number|complex]]. This is because many of the examples of reproducing kernel [[Hilbert space]]s are spaces of analytic functions. Also recall the sesquilinearity convention: the inner product is linear in the second variable.
Let ''X'' be an arbitrary [[set]] and ''H'' a [[Hilbert space]] of complex-valued functions on ''X''. ''H'' is a reproducing kernel Hilbert space iff the linear map
:<math> f \mapsto f(x) </math>
is [[norm-continuous]] for any elementfrom ''xH'' ofto ''X''the complex numbers is continuous. By the [[Riesz representation theorem]], this implies that there exists an element ''K''<sub>''x''</sub> of ''H'' such that for every function ''f'' in the space,
