I'm a CS student and I'm trying to learn RKHS theory to understand the passages made in this paper . Among the bibliography I'm using there are "On the mathematical fundamentals of learning" and "Learning with Kernels". I think I've got a good grasp of the relevant theory, but there's still something that bugs me.

As far as I understood a RKHS is a Hilbert space $H\subseteq \mathbb{C}^X$ where $X$ is a generic set of objects such that $H=\overline{span\{k_x|x\in X\}}$ and $\langle f,k_x\rangle=f(x)$ where $k_x=k(.,x)\;\forall x\in X$ and $k$ is a (Mercer) kernel. The inner product is defined as $\langle f,g\rangle=\int_X\int_X \alpha(x')\beta(x)k(x,x')dxdx'$ where $\alpha,\beta \in \mathbb{C}^X$, $f=\int_X\alpha(x')k(x,x')dx'$ and $g=\int_X\beta(x')k(x,x')dx'$. Let's call this "continuos-whole" definition.

However, given $D=\{x_1,x_2,...x_n\}\subset X$ a subspace $H_D\subset H$ could be defined by restricting the definition above to $H_D=\overline{span\{k_{x_i}|x_i\in D\}}=\{f\in \mathbb{C}^X|f=\overset{i=s}{\underset{i=1}{\sum}} \alpha_ik(x,x_i)\quad \alpha_i\in \mathbb{C} \;, r\in \mathbb{N}\}$ and thus define $\langle f,g\rangle_{H_D}=\overset{i=s}{\sum_{i=1}}\overset{j=s}{\sum_{j=1}}\alpha_i\beta_jk(x_i,x_j)$. Let this be the "discrete-finite" definition

Assuming this is correct, the references tend to define $H$ as both being spanned by $\{k_x|x \in X\}$ and having inner product $\langle .,.\rangle_{H_D}$, and that would be possible if and only if $\overline{span\{k_x|x\in X\}}=\overline{span\{k_{x_i}|x_i\in D\}}$ which seems bogus to me.

Then, regarding the article, there's a passage of which I'm unsure. Let $H$ be a RKHS, i.e. a Hilbert space s.t. every point-evaluation functional is bounded, which would entail a "whole-continuous" definition. Then $H$ can be orthogonal decomposed in $H=H_D\oplus H_D^\bot$ where $H_D^\bot \bot H_D \rightarrow \langle f,g\rangle=0 \forall f\in H_D \;\wedge\; g\in H_D^\bot$. The paper says that $H_D^\bot=\{g|g(x_i)=0\forall x_i\in D\}$. Starting from the definition of orthogonality, I carried out the following proof: $\forall f \in H_D \;\wedge\; g \in H_D^\top$ $\langle f,g \rangle=\int_X\int_X \alpha(x')\beta(x)k(x,x')dxdx'=\int_X \beta(x)f(x)dx$ $=\sum_{i \in 1..s}\alpha_i\int_X \beta(x)k(x,x_i)dx=\sum_{i \in 1..s} \alpha_i g(x_i)=0 \forall \alpha_i \in \mathbb{C}\rightarrow g(x_i)=0 \forall i\in 1..s$, for which I applied the reproducing property and the definiton of $g$ in this order. is this correct?