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 Nachman Aronszajn (1907-1980) and Stephan Bergman (1895-1987) in 1950. Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications, it is a topological space or a vector space or both. ... In mathematics, the operator norm is a norm defined on the space of bounded operators between two Banach spaces. ... Year 1950 (MCML) was a common year starting on Sunday (link will display the full calendar) of the Gregorian calendar. ...

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. The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ...

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 In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation...

from H to the complex numbers is continuous for any x in X. By the Riesz representation theorem, this implies that for given x there exists an element K_x of H with the property that: There are several well-known theorems in functional analysis known as the Riesz representation theorem. ...

The function

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

For example, when X is finite and H consists of all complex-valued functions on X, then an element of H can be represented as an array of complex numbers. If the usual inner product is used, then K_x is the function whose value is 1 at x and 0 everywhere else.

In other contexts, (*) amounts to saying

for every f, where X is often the real numbers or Rⁿ.

Bergman kernel

The Bergman kernel is defined for open sets D in Cⁿ. Take the Hilbert H space of square-integrable functions, for the Lebesgue measure on D, that are holomorphic functions. The theory is non-trivial in such cases as there are such functions, which are not identically zero. Then H is a reproducing kernel space, with kernel function the Bergman kernel; this example, with n = 1, was introduced by Bergman in 1922. In mathematics, the term integrable function refers to a function whose integral may be calculated. ... In mathematics, the Lebesgue measure is the standard way of assigning a length, area or volume to subsets of Euclidean space. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...

Moore-Aronszajn theorem

Given a positive definite kernel K, we can construct a unique RKHS H with K as the reproducing kernel. In operator theory, a positive definite kernel is a generalization of a positive matrix. ...

Define the operator by

or, equivalently, (where denotes the standard inner product on L₂(X)). Let H be the image of L₂ (X) under T_K and define an inner product on H by

Note that T_K^-1 is self-adjoint (we will write down exactly what it is later) and so . It is now easy to check that this defines a reproducing kernel Hilbert space. Indeed,

as required.

Mercer's theorem gives us another way to represent H. Let {λ_i} be a sequence of eigenvalues of T_K and let {e_i} be the corresponding eigenvectors. Then we can write the operator T_K as In mathematics and functional analysis Mercers theorem is a representation of a symmetric positive-definite function on a square as a sum of a convergent sequence of product functions. ...

and we can write the inner product on H as

.

See also

• Positive definite kernel
• Mercer's theorem
• Kernel trick

In operator theory, a positive definite kernel is a generalization of a positive matrix. ... In mathematics and functional analysis Mercers theorem is a representation of a symmetric positive-definite function on a square as a sum of a convergent sequence of product functions. ... In machine learning, the kernel trick is a method for converting a linear classifier algorithm into a non-linear one by using a non-linear function to map the original observations into a higher-dimensional space; this makes a linear classification in the new space equivalent to non-linear classification...

References

• Nachman Aronszajn, Theory of Reproducing Kernels, Transactions of the American Mathematical Society, volume 68, number 3, pages 337-404, 1950.