Given a Hilbert space H and an element x0 in H, we can define a map φ_{x0}(x) on H in terms of the inner product between x and x0. This map is linear and bounded (in some sense). The Riesz Representation Theorem claims that the converse is true. That is, any linear map (functional) in H can be uniquely represented as an inner product. As a consequence, function application (such as f(x)) can be represented as the inner product between the function and a reproducing kernel.
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