Paley–Wiener integral

In mathematics, the Paley–Wiener integral is a simple . When applied to classical Wiener space, it is less general than the Itō integral, but the two agree when they are both defined.

The integral is named after its discoverers, Raymond Paley and Norbert Wiener.

Let i : H → E be an abstract Wiener space with abstract Wiener measure γ on E. Let j : E^∗ → H be the adjoint of i. (We have abused notation slightly: strictly speaking, j : E^∗ → H^∗, but since H is a Hilbert space, it is isometrically isomorphic to its dual space H^∗, by the Riesz representation theorem.)

It can be shown that j is an injective function and has dense image in H. Furthermore, it can be shown that every linear functional f ∈ E^∗ is also square-integrable: in fact,

This defines a natural linear map from j(E^∗) to L²(E, γ; R), under which j(f) ∈ j(E^∗) ⊆ H goes to the equivalence class [f] of f in L²(E, γ; R). This is well-defined since j is injective. This map is an isometry, so it is continuous.

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