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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     author = {A. V. Uglanov},
     title = {The {Newton--Leibniz} formula on {Banach} spaces and approximation of functions of an infinite-dimensional argument},
     journal = {Izvestiya. Mathematics },
     pages = {145--161},
     publisher = {mathdoc},
     volume = {30},
     number = {1},
     year = {1988},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1988_30_1_a7/}
}
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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/