Notion of generalized function/distribution for functional derivatives? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T19:38:40Z http://mathoverflow.net/feeds/question/56223 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/56223/notion-of-generalized-function-distribution-for-functional-derivatives Notion of generalized function/distribution for functional derivatives? Will 2011-02-21T22:16:25Z 2012-09-17T11:23:42Z <p>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.</p> http://mathoverflow.net/questions/56223/notion-of-generalized-function-distribution-for-functional-derivatives/107373#107373 Answer by jbc for Notion of generalized function/distribution for functional derivatives? jbc 2012-09-17T10:48:43Z 2012-09-17T10:48:43Z <p>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".</p> http://mathoverflow.net/questions/56223/notion-of-generalized-function-distribution-for-functional-derivatives/107376#107376 Answer by AlexArvanitakis for Notion of generalized function/distribution for functional derivatives? AlexArvanitakis 2012-09-17T11:23:42Z 2012-09-17T11:23:42Z <p>Yes there is a notion of generalised function in infinite dimensions, see <a href="http://www.encyclopediaofmath.org/index.php/White_noise_analysis" rel="nofollow">http://www.encyclopediaofmath.org/index.php/White_noise_analysis</a> and the references contained therein. These functions are called "Hida distributions".</p>