Suppose I have a RKHS $\mathcal{H}$ on a set $X$ with kernel $K$. Moreover, othere are $\{x_1,\dots,x_n\}\subset X$ distinct points. We denote by $Q=(K(x_i,x_j))\in\mathbb{R}^{n\times n}$. In An Introduction to the Theory of Reproducing Kernel Hilbert Spaces, by V.I. Paulsen and M. Raghupathi they say for each $f\in\mathcal{H}$ there is a vector $w\in\mathbb{C}^n$ such that
$$Qw = (f(x_1),\dots,f(x_n)) $$
How can this conclusion be drawn? I was trying to work with the reproducing kernel property but didn't manage to get.
$$f(x_i) = \langle f, K(\cdot, x_i)\rangle $$
and trying to invoke the adjoint, without any success. Any hint / help would be much appreciated
Define $k_i(x):=K(x,x_i)$ for $i=j,\ldots,n$. Project $\bar f\in\mathscr{H}$ orthogonally onto the span of the $k_i$ to obtain $$ \bar f=\sum_{i=1}^n \alpha_i k_i + g$$ for suitable $\alpha_i\in\mathbb{C}$ and a $g\in\mathscr{H}$ orthogonal to the span. Now for all $i=1,\ldots,n$ $$ \bar f(x_j)=\langle \bar f,k_j\rangle = \sum_{i=1}^n \alpha_i \langle k_i, k_j\rangle=\sum_{i=1}^n \alpha_i K_{ji}$$ where we used the fact that $\langle g, k_j\rangle=0$ due to orthogonality.
With $v=\big(f(x_1),\ldots, f(x_n)\big)$, $w=\bar \alpha$ and $\bar Q=K(x_j,x_i)$ this is $$ \bar v_j = \big( \bar Q\alpha\big)_j$$ or $$ v=Qw.$$