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Digging in the definition of Reproducing Kernel Hilbert Spaces (RKHS) I came across the following example taken from pages 49-51 of [1]:

Given the kernel $k(x,y) = \langle x,y\rangle^2$, with $x,y\in \mathbb R^2$, one can show that both

$\phi(x) = [x_1^2,~ x_2^2,~ x_1x_2, ~x_1x_2]$, with feature space $\mathcal H = \mathbb R^4$

$\phi^{\prime}(x) = [x_1^2,~ x_2^2,~ \sqrt 2 x_1x_2]$, with feature space $\mathcal H^\prime = \mathbb R^3$

are valid feature maps for the kernel. However, the slides point out that neither $\mathcal H$ nor $\mathcal H^\prime$ are RKHS because the RKHS must by definition be unique for a given kernel. Can you explain what the RKHS would be in this case?

Other references on the topic state that the RKHS $\mathcal H_k$ is unique up to isomorphism [2]. Under this definition I guess both $\mathcal H$ and $\mathcal H^\prime$ can be considered RKHS by being isomorphic to $\mathcal H_k$.

[1] http://www.stats.ox.ac.uk/~sejdinov/teaching/atml14/Theory_slides1_2014.pdf

[2] http://users.umiacs.umd.edu/~hal/docs/daume04rkhs.pdf

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