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For any Lie supergroup whose underlying Lie group is reductive, we prove anextension of the Casselman-Wallach globalisation theorem: There is anequivalence between the category of Harish-Chandra modules and the category ofSF-representations (smooth Fr\'echet representations of moderate growth) whosemodule of finite vectors is Harish-Chandra. As an application, we extend to Liesupergroups a general general form of the Gel'fand-Kazhdan criterion due toSun-Zhu.

Fr\'echet globalisations of Harish-Chandra supermodules