Let(π_{λ}, ℋ_{λ}) be a unitary highest weight representation ofthe connected Lie group G and ℊ its Lie algebra. Thenℊ contains an invariant closed convex cone W_{\rm{max}}such that, for each X∈W_{\rm{max}}^0, the selfadjoint operatori·dπ_{λ}(X) is bounded from above. We show that for each suchX, the space ℋ_{λ}^{∞} of smooth vectors forthe action of G on ℋ_{λ} coincides with the set𝒟^{∞}(dπ_{λ}(X)) of smooth vectors for the generally unboundedoperator dπ_{λ}(X).