Which Spaces can be Embedded in Reproducing Kernel Hilbert Spaces?

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 \emph{proportional} to the dimension $d$ in order to allow for an intermediate reproducing kernel Hilbert space $H$.