## Bulletin of the Belgian Mathematical Society - Simon Stevin

### The convenient setting for ultradifferentiable mappings of Beurling- and Roumieu-type defined by a weight matrix

Gerhard Schindl

#### Abstract

We prove in a uniform way that all ultradifferentiable function classes $\mathcal{E}_{\{\mathcal{M}\}}$ of Roumieu-type and $\mathcal{E}_{(\mathcal{M})}$ of Beurling-type defined in terms of a weight matrix $\mathcal{M}$ admit a convenient setting if $\mathcal{M}$ satisfies some mild regularity conditions. For $\mathcal{C}$ denoting either $\mathcal{E}_{\{\mathcal{M}\}}$ or $\mathcal{E}_{(\mathcal{M})}$ the category $\mathcal{C}$ is cartesian closed, i.e. $\mathcal{C}(E\times F,G)\cong\mathcal{C}(E,\mathcal{C}(F,G))$ for $E,F,G$ convenient vector spaces. As special cases one obtains the classes $\mathcal{E}_{\{M\}}$ and $\mathcal{E}_{(M)}$ respectively $\mathcal{E}_{\{\omega\}}$ and $\mathcal{E}_{(\omega)}$ defined by a weight sequence $M$ respectively a weight function $\omega$.

#### Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 22, Number 3 (2015), 471-510.

Dates
First available in Project Euclid: 16 September 2015

https://projecteuclid.org/euclid.bbms/1442364593

Digital Object Identifier
doi:10.36045/bbms/1442364593

Mathematical Reviews number (MathSciNet)
MR3396997

Zentralblatt MATH identifier
1337.46020

#### Citation

Schindl, Gerhard. The convenient setting for ultradifferentiable mappings of Beurling- and Roumieu-type defined by a weight matrix. Bull. Belg. Math. Soc. Simon Stevin 22 (2015), no. 3, 471--510. doi:10.36045/bbms/1442364593. https://projecteuclid.org/euclid.bbms/1442364593