A hopefully relatively simple question: is countable infinity an element of the set of natural numbers? A related question here has been answered with the statement that (classic) infinity is not part of the natural numbers, but what about countable infinity?
Context for this question: A definition of a Reproducing Kernel Hilbert Space (RKHS) I have come across is as follows:
The reproducing kernel Hilbert space (RKHS) $\mathcal{H}$ is the closure of the linear span $\{f:f(x)=\sum_{i=1}^{m} a_ik(x,x_i), \space a_i \in \mathbb{R}, \space m \in \mathbb{N}, x_i \in \mathcal{X}\}$ equipped with inner products $\left< f,g \right>_{\mathcal{H}}=\sum_{ij}a_ib_jk(x_i,x_j)$ for $g(x)=\sum_{i=1}^{m} b_ik(x,x_i)$.
Many examples or RKHSs have infinitely many dimensions, so we would require $m=\infty \in \mathbb{N}$. Feel free to ignore this context if it is not required for your answer.
No, you do not need $m = \infty$. Note that you are taking the closure of the linear span, which takes care of the elements requiring an infinite sum for their representation.