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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2$\begingroup$ 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. $\endgroup$– Nate EldredgeCommented Feb 22, 2011 at 5:25
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3$\begingroup$ 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. $\endgroup$– Zen HarperCommented Feb 22, 2011 at 8:01
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2 Answers
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There is a substantial literature on spaces of distributions on infinite dimensional spaces, in particular function spaces, so much so that there is a subsection of the AMS classification scheme devoted to it (35R15). This might be relevant for your question. Related subjects can be found in 46G. This work is mainly motivated by questions in quantum theory, especially quantum field theory. A possible starting point would be the work of Y.M. Berezanskii, e.g., his monograph "Self-adjoint operators in spaces of functions in infinitely many dimensiona".
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Yes there is a notion of generalised function in infinite dimensions, see http://www.encyclopediaofmath.org/index.php/White_noise_analysis and the references contained therein. These functions are called "Hida distributions".
