Given a Banach space E consisting of functions, we ask whether there exists a reproducing kernel Hilbert space H with bounded kernel such that \(E\subset H\) . More generally, we consider the question, whether for a given Banach space consisting of functions F with \(E\subset F\) , there exists an intermediate reproducing kernel Hilbert space \(E\subset H\subset F\) . We provide both sufficient and necessary conditions for this to hold. Moreover, we show that for typical classes of function spaces described by smoothness there is a strong dependence on the underlying dimension: the smoothness s required for the space E needs to grow proportional to the dimension d in order to allow for an intermediate reproducing kernel Hilbert space H.
