Reproducing kernel hilbert space

Gaussian kernel matrix can be factorized into $(\Phi \text{X}{)}^{\text{H}}\Phi \text{X}={\text{X}}^{\text{H}}{\Phi }^{\text{H}}\Phi \text{X}={\text{X}}^{\text{H}}\text{X}$, where $\Phi$ is Gaussian kernel basis matrix and $\text{X}$ is coefficients matrix of reproducing kernel Hilbert space $K(\cdot ,x)\in H_K$ https://www.jkangpathology.com/post/reproducing-kernel-hilbert-space/. A matrix is a system. A system takes input and gives output. A matrix is a linear system. Differentiation and Integration are linear systems. Fourier transformation matches input basis and operator (differentiation) basis.

Finally arrive at reproducing kernel Hilbert space. https://nzer0.github.io/reproducing-kernel-hilbert-space.html The above post introduces RKHS in Korean. It was helpful. I had struggled to understand some concepts in RKHS. What does mean Hilbert space in terms of feature expansion? ($f:X\to R$, $f\in H_K$) It was confusing the difference between $f$ and $f\left(x\right)$. $f$ means the function in Hilbert space and $f\left(x\right)$ is evaluation. I thought that the function can be represented by the inner product of the basis of feature space $K(\cdot ,x)$ and coefficients $f$, and the coefficients are vectors in feature space.

The reproducing kernel hilbert space (RKHS) was my motivation to study analysis. The hilbert space is a orthogonal normed vector space. I still do not know about the meaning of “reproducing kernal”. The RKHS appeared in the book titled An Introduction to Statisitical Learning written by Hastie. I began to google the meaning of the spaces such as the Hilbert, Banarch. I decided to read the Understaing Analysis written by Abbott.