I have a manifold $X$ and a reproducing kernel hilbert space $H$ of functions $X\rightarrow\mathbb{C}$. With kernel function $K:X\times X \rightarrow \mathbb{C}$, so $K(\cdot,x)$ is an element of $H$.
I want to show, that the dual function (is this the right term?) to the Kernel function is essentially the inner product in the Hilbert space. $$ K(\cdot,x)^*(\operatorname{D}K(\cdot,x)e_i) =\langle K(\cdot,x),\operatorname{D}K(\cdot,x)e_i\rangle. $$ ( $e_i$ is a basic vector of $T_xX$)
I know the Riesz representation theorem guarantees the existence of an element $f\in H$, s.t. $$ K(\cdot,x)^*(\operatorname{D}K(\cdot,x)e_i) =\langle f,\operatorname{D}K(\cdot,x)e_i\rangle.$$ So why would $f= K(\cdot,x)$?
I was thinking maybe because this is always the kernel function we get the desired formula. But I couldn't follow through with that argument in a rigorous way, so I suspect I still have some gaps in my understanding of the kernel function. I have been reading about it, but this has yet remained a mystery to me.