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This is a closed subset of the space of continuous functions with the supremum norm, by an apparently not so well-known lemma of Henry (see Proposition 3).
IfMis non-compact, then we define the weighted counting functionNM, w(T) via the inverse Laplace transform from which one can express the weighted counting function in terms of spectral data associated toM(see Proposition 5.1 and Remark 5.2).
and ; see Proposition 3.1.
See Proposition 1.
See Proposition 2.1 of [23].
See Proposition 2 in [10].
Proposition 53 (see Proposition 27 in [22]).
See Proposition A.1 in the Appendix.
Since, we get, see Proposition 2.2.
See Proposition 6.1 in Keller-Ressel et al. (2013).
When, then (see Proposition 7.3), and we get the following.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com