Reproducing kernel Hilbert space: Difference between revisions

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In this article we assume that Hilbert spaces are 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 element ''x'' of ''X''. By the [[Riesz representation theorem]], this implies that there exists an element ''K''_''x of ''H'' such that

:<math> f(x) = \langle K_x, f \rangle </math>

The function

:<math> K(x,y) = K_x(y) </math>

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

:<math> f(x) = \langle K(x, \cdot), f(\cdot)) \rangle </math>