I'm following the gentle introduction to Reproducing Kernel Hilbert Spaces From Zero to Reproducing Kernel Hilbert Spaces in Twelve Pages or Less by Hal Daumé III. I believe the author fully intended to hand-wave in several places, and that's been mostly fine for my purposes since I've been able to extract the main idea and fill in some of the details, but I do not have a slight idea about the following.

We pick it up in Section 6.4 on page 9 and follow the author's notation. Given a positive definite kernel $K$ over $X$, we wish to construct an RKHS $\mathcal{H}$ of functions $f : X \to \mathbb{R}$. For this construction, we need an inner product, so the author writes the obvious thing

$$ \langle k_x , k_y \rangle_{\mathcal{H}_K} = \Big\langle \sum_i \alpha_i k_{x_{i}}, \sum_i \beta_i k_{y_{i}} \Big\rangle_{X} $$

since we have already added the span of $\{k_x\}_{x\in X}$ to the RKHS under construction.

One page later, the author appeals to Mercer-Hilbert-Schmit theorems and Fourier analysis to argue that the dot product can be defined by

$$ \langle y, y^{\prime}\rangle_{\mathcal{H}_{K}} = \sum_{i=0}^{\infty} \dfrac{y_i y_{i}^{\prime}}{\lambda_i} $$

where $\lambda_i$ are eigenvalues and

$$ y_i := \int y(x) \phi_i (x) dx $$

for eigenfunctions $\phi$ (the decision to switch to $y\in \mathcal{H}$ was the author's).

**My question is NOT the derivation of this, but why it is needed**. What's so wrong with the first inner product we defined that forces us to come up with this second representation based on eigenfunctions? Thanks.