I'm a bit confused with the canonical feature map $\phi_{\mathrm{can}}(x)=k(x, \cdot)$ and explicit feature map $\phi_{\exp }(x)$.
Take the polynomial kernel $k\left(x, x^{\prime}\right)=\left(1+\left\langle x, x^{\prime}\right\rangle\right)^{2}$ as an example the canonical feature map lives in a function space, whereas the explicit feature map $\phi_{\exp }(x)=\left[1, \sqrt{2} x_{1}, \sqrt{2} x_{2}, x_{1}^{2}, \sqrt{2} x_{1} x_{2}, x_{2}^{2}\right]^{\top}$ is a vector in $\mathbb{R}^{6}$, not a function.
I'm wondering what's the connection between the vector space and function space above. Since every kernel has a unique RKHS, this polynomial kernel should have only one RKHS. Also, I don't understand how to represent a function in the vector space when the explicit feature is selected.
Any hint would be appreciated! Thanks a lot in advance.