Reproducing Kernel Hilbert Space

AnalysisFunctionalAnalysisPDEs A Reproducing Kernel Hilbert Space (RKHS) is a Hilbert Space of functions on such that the evaluation functional is in the topological dual space. In other words, the evaluation functional is a continuous (bounded), linear functional, given , : Now by the Riesz-Representation Theorem, every linear functional can be represented by an inner product with a unique element applying this to itself defines the reproducing kernel clearly this is a p.d.s. kernel The Moore-Aronszajn Theorem serves as a converse to this: every p.d.s. kernel defines a unique RKHS with itself as the reproducing kernel.