Geodesic
The Newton--Leibniz formula on Banach spaces and approximation of functions of an infinite-dimensional argument
Izvestiya. Mathematics , Tome 30 (1988) no. 1, pp. 145-161.

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Properties of integrals over infinite-dimensional nonlinear manifolds are analyzed. A certain double averaging operation is introduced for functions on abstract separable Banach spaces; this operation leads to uniform approximation by smooth (in the Fréchet sense) functions in the case of spaces (and certain other cases). Bibliography: 10 titles. 
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A. V. Uglanov. The Newton--Leibniz formula on Banach spaces and approximation of functions of an infinite-dimensional argument. Izvestiya. Mathematics , Tome 30 (1988) no. 1, pp. 145-161. http://geodesic.mathdoc.fr/item/IM2_1988_30_1_a7/

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