I'd like to know why inner products in Reproducing kernel Hilbert spaces are (linear) evaluation functionals.

I understand the Riesz representation theorem (in chapter 2 of "Principles of Functional Analysis" by Martin Schechter), and that inner products are linear functionals, and I know what an evaluation functional is; I just can't explain why an inner product (in a RKHS) is evaluation functional, and vise-versa.

Edit: to make things clearer, I'm aware that given a Hilbert space $\mathcal{H}$ having the inner-product $\langle x \;,\; y \rangle $, where $x , y \in \mathcal{H}$; if $y$ is fixed $\langle x \;,\; y \rangle $ assigns to each $x$ a number. What I don't understand is why the the relation

$\mathcal{F}_y(x)\;=\;\langle \; x \;,\;y\;\rangle$

appears to be sufficient to make the inner product a Dirac evaluation functional. Ad I said, I know that the inner product is a functional but I can't explain why it's a (Dirac) evaluation functional.

PS: I've posted a similar question on math.stackexchange but didn't get any replies.