Partial Derivative Approach to the Integral Transform for the Function Space in the Banach Algebra

We investigate some relationships among the integral transform, the function space integral and the first variation of the partial derivative approach in the Banach algebra defined on the function space. We prove that the function space integral and the integral transform of the partial derivative in some Banach algebra can be expanded as the limit of a sequence of function space integrals.


Introduction
The first variation defined by the partial derivative approach was defined in [1]. Relationships among the Function space integral and transformations and translations were developed in [2][3][4]. Integral transforms for the function space were expanded upon in [5][6][7][8][9].
A change of scale formula and a scale factor for the Wiener integral were expanded in [10][11][12] and in [13] and in [14].
Relationships among the function space integral and the integral transform and the first variation were expanded in [13,15,16] and in [17,18] In this paper, we expand those relationships among the function space integral, the integral transform and the first variation into the Banach algebra [19].

Preliminaries
Let

F(x) dm(x)
is called the Wiener integral of a function F defined on the function space C 0 [0, T].
A subset E of C 0 [0, T] is said to be scale-invariant measurable provided ρE ∈ M for all ρ > 0 and a scale invariant measurable set N is said to be scale-invariant null provided m(ρN) = 0 for each ρ > 0. A property that holds except on a scale-invariant null set is said to hold scale-invariant almost everywhere (s-a.e.). If two functions F and G are equal s-a.e., we write F ≈ G.
Definition 3 (Ref. [1]). The first variation of a Wiener measurable functional F in the direction w ∈ C 0 [0, T] which is defined by the partial derivative as We will denote it by [D, F, x, w].

Remark 1.
For a ∈ C + and b ∈ R,

Proof. First we have
Therefore we have using Lemma 4.