Notion of generalized function/distribution for functional derivatives?

Is there any work on defining something analogous to a generalized function for functions whose domain is a Hilbert or Banach space? Is there an extension of the notion of Frechet/Hadamard/Gateaux derivatives that somehow resembles the theory of distributions? I am hoping to take something like a Taylor expansion of a non-differentiable function from a Hilbert space to the real line.

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Interesting question. I don't know anything about this. I guess the main obstruction is that distribution theory relies on integration by parts, with respect to a translation-invariant measure whose "derivative" vanishes, whereas in infinite dimensions no such measure exists. There are derivatives as you mention, and there are measures such as Gaussian ones, but without the translation invariance it doesn't seem that this would give a satisfactory theory. –  Nate Eldredge Feb 22 '11 at 5:25
Maybe it would be helpful to give more details of a concrete question you want to solve with such a theory, so we know what you're aiming for. –  Zen Harper Feb 22 '11 at 8:01