This note consists of two largely independent parts. In the first part we give conditions on the kernel $k: \Omega \times \Omega \rightarrow \mathbb{R}$ of a reproducing kernel Hilbert space $H$ continuously embedded via the identity mapping into $L^2(\Omega, \mu),$ which are equivalent to the fact that $H$ is even compactly embedded into $L^2(\Omega, \mu).$ 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 $L^2(\Omega, \mu)$ is compact. We provide a simple criterion for checking compactness of the embedding of a reproducing kernel Hilbert space with the kernel given by $$\sum_{u \in \mathcal{U}} \gamma_u \bigotimes_{j \in u}k,$$ where $\mathcal{U} = \{u \subset \mathbb{N}: |u| < \infty\},$ and $(\gamma_u)_{u \in \mathcal{U}}$ is a sequence of non-negative numbers, into an appropriate $L^2$ space.

翻译：本注释由两个基本上独立的部分组成。 在第一部分, 我们给出了对内核的条件 :\ Omega\ times\ Omega\ times\ Omega\rightrow\ mathb{R} 复制内核Hilbert 空间$H$, 通过身份映射持续嵌入 $L2\\\ Omega,\mu), 这相当于 $ 美元甚至被压缩嵌入 $L2\\ Omega,\ mu。 在第二部分, 我们考虑一个无限变值$$$2$ 的假设 。 假设的是, 复制内核1+k$ 的内核 Hilbert 空间的再生产内核 $ $2\\\ mega,\ mu) 的顺序很紧凑 。 我们提供了一个简单的标准, 用来检查再生产内核空间空间与由美元\ sumanu\ sum $\ mathal $ $U\\ gamma_\\\ nu= nu a_ nu= nu= nu= nu= number. $_ u\\\\\\\\\\\ number a_\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\