Open Access
2018 Banach Spaces for the Schwartz Distributions
Tepper L. Gill
Real Anal. Exchange 43(1): 1-36 (2018). DOI: 10.14321/realanalexch.43.1.0001


This paper is a survey of a new family of Banach spaces \(\mathcal{B}\) that provide the same structure for the Henstock-Kurzweil (HK) integrable functions as the \(L^p\) spaces provide for the Lebesgue integrable functions. These spaces also contain the wide sense Denjoy integrable functions. They were first use to provide the foundations for the Feynman formulation of quantum mechanics. It has recently been observed that these spaces contain the test functions \(\mathcal{D}\) as a continuous dense embedding. Thus, by the Hahn-Banach theorem, \(\mathcal{D}' \subset \mathcal{B}'\). A new family that extend the space of functions of bounded mean oscillation \(BMO[\mathbb{R}^n]\), to include the HK-integrable functions are also introduced.


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Tepper L. Gill. "Banach Spaces for the Schwartz Distributions." Real Anal. Exchange 43 (1) 1 - 36, 2018.


Published: 2018
First available in Project Euclid: 2 May 2018

zbMATH: 06924870
MathSciNet: MR3816428
Digital Object Identifier: 10.14321/realanalexch.43.1.0001

Primary: 46
Secondary: 47A16

Keywords: Banach space , Feynman path integral , Henstock-Kurzweil integral , Markov processes , Navier-Stokes

Rights: Copyright © 2018 Michigan State University Press

Vol.43 • No. 1 • 2018
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