A short note on compact embeddings of reproducing kernel Hilbert spaces in $L^2$ for infinite-variate function approximation

This note consists of two largely independent parts. In the first part we give conditions on the kernel $k:\mathrm{\Omega }\times \mathrm{\Omega }\to \mathbb{R}$ of a reproducing kernel Hilbert space $H$ continuously embedded via the identity mapping into $L^2(\mathrm{\Omega },\mu ),$ which are equivalent to the fact that $H$ is even compactly embedded into $L^2(\mathrm{\Omega },\mu )\mathrm{.}$ In the second part we consider a scenario from infinite-variate $L^2$-approximation. Suppose that the embedding of a reproducing kernel Hilbert space of univariate functions with reproducing kernel $1+k$ into $L^2(\mathrm{\Omega },\mu )$ is compact. We provide a simple criterion for checking compactness of the embedding of a reproducing kernel Hilbert space with the kernel given by $$\sum _{u\in \mathcal{U}}{\gamma }_u\underset{j\in u}{⨂}k,$$ where $\mathcal{U}=\{u\subset \mathbb{N}:\mathrm{\mid }u\mathrm{\mid }<\mathrm{\infty }\},$ and $({\gamma }_u{)}_{u\in \mathcal{U}}$ is a sequence of non-negative numbers, into an appropriate $L^2$ space.