Riesz Representation Theorem for G-Integrable Distributions

• Main Author: Gealone, Aries
• Format: text
• Published: Archīum Ateneo 2020
• Subjects: Riesz representation theorem, Henstock integral.
• Online Access: https://archium.ateneo.edu/theses-dissertations/532
• Institution: Ateneo De Manila University

Let the set of functions defined on R with support contained in [a, b] and has deriva- tives of all orders be denoted by D(a, b). Let the space of continuous linear functionals (distributions) defined on D(a, b) be denoted by D0 (a, b). In the sense of distribution, every distribution has a primitive and a derivative (Griffel, 1981). Denote the space of Henstock-integrable functions on [a, b] by E(a, b). Each element of E(a, b) generates a distribution in D0 (a, b). The Banach space G(a, b) ⊂ D0 (a, b) is the completion of the space of distributions generated by each element of E(a, b), with respect to the G-norm defined as kfkG = sup {|F(x)| : a ≤ x ≤ b, F0 = f in the sense of distribution, and F ∈ C0[a, b]} . D.D. Ang, P.Y. Lee, and L.K. Vy (1990) developed an integration on G(a, b) which extends Henstock integration from functions on R to distributions. This study discusses the development of the Riesz Representation theorem for G-integrable distributions on a non-degenerate closed interval [a, b], which involves Henstock integration and distri- bution theory.