Before revision:   The manuscript concludes with the description of a Dynamic Mode Decomposition algorithm that converges when a dense collection of occupation kernels, arising from the data, are leveraged in the analysis.
After revision:  However, in the case where the domain is embedded in the range, an eigenfunction approach is still achievable, where a more typical DMD routine is established, but that leverages a finite rank representation that converges in norm. The manuscript concludes with the description of a Dynamic Mode Decomposition algorithm that converges when a dense collection of occupation kernels, arising from the data, are leveraged in the analysis.