Consider $\mathcal{X}=\mathbb{R}$, and $k(x,y)=xy=[\frac{x}{\sqrt{2}},\frac{x}{\sqrt{2}}]\cdot [\frac{y}{\sqrt{2}},\frac{y}{\sqrt{2}}]^T$, where we thus can define two kinds of feature maps $\Phi_1(x)=x$ and $\Phi_2(x)=[\frac{x}{\sqrt{2}},\frac{x}{\sqrt{2}}]$, and the feature space are respectively $\mathcal{H}_1=\mathbb{R}$ and $\mathcal{H}_2=\mathbb{R}^2$.
Then a claim says that: neither of these two feature space is a RKHS ( Reproducing Kernel Hilbert Space), as they are not spaces of functions on $\mathcal{X}=\mathbb{R}$.
Could anyone help to interpret this claim? I still do not know how to prove if they are in RKHS or not.