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Implicit functions in finite corank on the Wiener space. (English) Zbl 0546.60003
Stochastic analysis, Proc. Taniguchi Int. Symp., Katata & Kyoto/Jap. 1982, North-Holland Math. Libr. 32, 369-386 (1984).
[For the entire collection see Zbl 0538.00017.]
The author develops the foundations of a differential calculus on Wiener space. Unfortunately Wiener functionals are not continuous for any Banach norm. A certain scale of capacities is introduced and the concept of a slim set defined, that is a set of capacity zero for all the scale of capacities. A smooth function can then be redefined outside a slim set, and it is then continuous (together with its derivatives) relative to the Banach space norm. It is shown that the projection of a slim set by a linear projection of finite corank is a slim set on the image. Using these concepts an implicit function theorem is proved in the final section.
Reviewer: R.J.Elliot

MSC:
60B05 Probability measures on topological spaces
60J65 Brownian motion
60J45 Probabilistic potential theory
60H25 Random operators and equations (aspects of stochastic analysis)