Integration By Parts Operator - Definition

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 (fixed) subset of the class of functions defined on E. A linear operator A : SL2(E, μ; R) is said to be an integration by parts operator for μ if

for every C1 function φ : ER and all hS for which either side of the above equality makes sense. In the above, Dφ(x) denotes the Fréchet derivative of φ at x.

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