Reproducing kernel Hilbert space

In functional analysis (a branch of mathematics), a reproducing kernel Hilbert space is a function space in which pointwise evaluation is a continuous linear functional. Equivalently, they are spaces that can be defined by reproducing kernels. The subject was originally and simultaneously developed by N. Aronszajn and S. Bergman in 1950.

In this article we assume that Hilbert spaces are complex. This is because many of the examples of reproducing kernel Hilbert spaces 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

${\displaystyle f\mapsto f(x)}$

is norm-continuous for any element x of X. By the Riesz representation theorem, this implies that there exists an element K_x of H such that for every function f in the space,

${\displaystyle f(x)=\langle K_{x},f\rangle .}$

The function

${\displaystyle K(x,y)=K_{x}(y)}$

is called a reproducing kernel for the Hilbert space. In fact, K is uniquely determined by the condition

${\displaystyle f(x)=\langle K(x,\cdot ),f(\cdot ))\rangle }$

for every f. In some concrete contexts this amounts to saying

${\displaystyle f(x)=\int _{\Omega }K(x,y)f(y)\,dy}$

for every f, where Ω is the appropriate ___domain, often the real numbers or Rⁿ.