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Let $G$ be a connected and simply connected Banach–Lie group. On the complex enveloping algebra of its Lie algebra $𝔤$ we define the concept of an analytic functional and show that every positive analytic functional $\lambda$ is integrable in the sense that it is of the form $\lambda \left(D\right)=\langle \mathtt{d}\pi \left(D\right)v,v\rangle$ for an analytic vector $v$ of a unitary representation of $G$. On the way to this result we derive criteria for the integrability of $*$-representations of infinite dimensional Lie algebras of unbounded operators to unitary group representations. ...