Reproducing kernel Hilbert space: Difference between revisions

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In [[mathematics]] and [[functional analysis]] a '''reproducing kernel [[Hilbert space]]''' is a [[function space]] in which pointwise evaluation is a continuous linear functional. Alternatively, we will show 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 spaces are complex. This is because many of the examples of reproducing kernel [[Hilbert spacesspace]]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 element ''x'' of ''X''. By the [[Riesz representation theorem]], this implies that there exists an element ''K''_''x of ''H'' such that