## Real Analysis Exchange

### Banach Spaces for the Schwartz Distributions

Tepper L. Gill

#### Abstract

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.

#### Article information

Source
Real Anal. Exchange, Volume 43, Number 1 (2018), 1-36.

Dates
First available in Project Euclid: 2 May 2018

https://projecteuclid.org/euclid.rae/1525226419

Digital Object Identifier
doi:10.14321/realanalexch.43.1.0001

Mathematical Reviews number (MathSciNet)
MR3816428

Zentralblatt MATH identifier
06924870

Subjects
Primary: 46
Secondary: 47A16: Cyclic vectors, hypercyclic and chaotic operators

#### Citation

Gill, Tepper L. Banach Spaces for the Schwartz Distributions. Real Anal. Exchange 43 (2018), no. 1, 1--36. doi:10.14321/realanalexch.43.1.0001. https://projecteuclid.org/euclid.rae/1525226419