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I am aware that there is a whole lot of literature out there on the expansive field of infinite-dimensional analysis. Moreover, almost every book I've glanced at so far seems to have a different focus than the next one. Specifically, I am interested in the study of maps $f: M \to H$, where $M$ is a finite-dimensional manifold and $H$ is a Hilbert space (or a Banach space). Here are some concrete topics that I want to learn more about

a) Fréchet-Derivatives (generalizing total derivatives), Gateaux-Derivaties (generalizing directional derivates), their connection, and the infinite-dimensional analogue of classic theorems such as the Mean-Value Theorem or Schwarz' theorem.

b) (This is most important to me) Differential forms over $M$ with values in $H$, the corresponding concept of exterior derivative on such forms, integrating $k$-forms over $k$-dimensional embedded submanifolds (via the Bochner Integral) and a generalization of Stokes Theorem.

Does anybody know a good reference for this ?

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You should check out Cartan's "Differential Forms" in that book the very first definition of differential forms deals with forms defined over an open subset of a Banach and with values in a Banach.

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