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I'm trying to understand reproducing kernel Hilbert spaces (RKHSs) from scientific papers, however I don't find any gentle introduction. However, my main problem, at the moment, seems to be to understand the used notation. I understand that a kernel $k$ is a function which maps features to the respective inner product in a Hilbert space, i.e. $k(x,z)=<\phi(x), \phi(z)>_H$ with $\phi$ mapping from the original space to an Hilbert space. However, what I don't understand is the $(\cdot)$ notation, e.g. $k(\cdot, z)$. What means the dot instead of the first argument? In several papers I read something like $k(\cdot, z)=\phi(z)$ but what is the meaning of this? Intuitively I would think something like this: $\forall x, k(x, z)=\phi(z)$, but it does not seem to make any sense to me... Also the following answer Reproducing Kernel Hilbert Space - notation and basics does not clarify my doubts... So, what is the real meaning of $\cdot$? Does anyone know any clear introduction on this topic?

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  • $\begingroup$ $k(\cdot,z)$ refers to the function $$x\mapsto k(x,z)$$ $\endgroup$
    – Ruy
    May 30, 2022 at 15:23
  • $\begingroup$ so $k(\cdot,z)= \phi(z) \Leftrightarrow \forall x, k(x,z) = \phi(z)$? But in this manner $k(x,z)$ is not $<\phi(x), \phi(z)>$ ... $\endgroup$
    – volperossa
    May 30, 2022 at 15:47
  • $\begingroup$ @volperossa: What Ruy is saying is that is $k(.,z)$ is just a short hand notation for $x \rightarrow k(x,z) $ so it just means $ < \phi(x), \phi(z) > $ for any $x$ rather than a specific $x$. $\endgroup$
    – mark leeds
    May 30, 2022 at 16:11
  • $\begingroup$ so what is the meaning of $k(\cdot, z)=\phi(z)$? $\endgroup$
    – volperossa
    May 30, 2022 at 17:51

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