# reproducing kernel hilbert space notation

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?

• $k(\cdot,z)$ refers to the function $$x\mapsto k(x,z)$$
– Ruy
May 30, 2022 at 15:23
• so $k(\cdot,z)= \phi(z) \Leftrightarrow \forall x, k(x,z) = \phi(z)$? But in this manner $k(x,z)$ is not $<\phi(x), \phi(z)>$ ... May 30, 2022 at 15:47
• @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 $x$. May 30, 2022 at 16:11
• so what is the meaning of $k(\cdot, z)=\phi(z)$? May 30, 2022 at 17:51