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Consider a positive definite kernel $K:\mathbb N\times \mathbb N\rightarrow \mathbb R$. Denote the unique RKHS associated with $K$ by $\mathcal H_K$. The RKHS $\mathcal H_K$ consists of \begin{align} \mathcal H_0= \left\{\sum_{t'=1}^n a_{t'}K(\cdot,t') \mid n\in\mathbb N, a_{t'}\in\mathbb R \right\} \end{align} and its completion. The completion of $\mathcal H_0$ is the set of functions which are pointwise limits of Cauchy sequences in $\mathcal H_0$ with the inner product defined by \begin{align} \langle h_a,h_b \rangle_{\mathcal H_K} = \sum_{t=1}^n\sum_{t'=1}^m a_{t}b_{t'}K(t,t') \end{align} where $h_a=\sum_{t=1}^n a_{t}K(t,\cdot)$ and $h_b=\sum_{t'=1}^m b_{t'}K(\cdot,t')$ with $a_{t},b_{t'}\in\mathbb R$.

My question is:

If $\mathcal H_K\subset \ell^1$, is $\{f_n\}$ with $f_n = \sum_{t=1}^n a_{t}K(\cdot,t)$ and $\{a_t\}\in\ell^\infty$ a Cauchy sequence in $\mathcal H_0$?

This problem may be a basic one (?) but has bothered me for a while. Any ideas and suggestions will be great. Thank you.

I have read some papers and books regarding RKHS, e.g.,

A. Berlinet and C. Thomas-Agnan, Reproducing kernel Hilbert spaces in probability and statistics. Kluwer Academic Boston, 2004.

However, I did not find relevant discussions.

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