Integration by parts operator
In mathematics, an integration by parts operator is a linear operator used to formulate integration by parts formulae; the most interesting examples of integration by parts operators occur in infinite-dimensional settings and find uses in stochastic analysis and its applications.Definition
Let E be a Banach space such that both E and its continuous dual space E∗ are separable spaces; let μ be a Borel measure on E. Let S be any subset of the class of functions defined on E. A linear operator A : S → L2 is said to be an integration by parts operator for μ if
for every C1 function φ : E → R and all h ∈ S for which either side of the above equality makes sense. In the above, Dφ denotes the Fréchet derivative of φ at x.Examples
