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An RKHS is associated with a kernel that reproduces every function in the space in the sense that for every <math>x</math> in the set on which the functions are defined, "evaluation at <math>x</math>" can be performed by taking an inner product with a function determined by the kernel. Such a ''reproducing kernel'' exists if and only if every evaluation functional is continuous.
The reproducing kernel was first introduced in the 1907 work of [[Stanisław Zaremba (mathematician)|Stanisław Zaremba]]{{fact}} concerning [[boundary value problem]]s for [[Harmonic function|harmonic]] and [[Biharmonic equation|biharmonic functions]]. [[James Mercer (mathematician)|James Mercer]] simultaneously examined [[Positive-definite kernel|functions]] which satisfy the reproducing property in the theory of [[integral equation]]s. The idea of the reproducing kernel remained untouched for nearly twenty years until it appeared in the dissertations of [[Gábor Szegő]], [[Stefan Bergman]], and [[Salomon Bochner]]. The subject was eventually systematically developed in the early 1950s by [[Nachman Aronszajn]] and Stefan Bergman.<ref>Okutmustur</ref>
These spaces have wide applications, including [[complex analysis]], [[harmonic analysis]], and [[quantum mechanics]]. Reproducing kernel Hilbert spaces are particularly important in the field of [[statistical learning theory]] because of the celebrated [[representer theorem]] which states that every function in an RKHS that minimises an empirical risk functional can be written as a [[linear combination]] of the kernel function evaluated at the training points. This is a practically useful result as it effectively simplifies the [[empirical risk minimization]] problem from an infinite dimensional to a finite dimensional optimization problem.
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