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

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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

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1442364593

Digital Object Identifier
doi:10.36045/bbms/1442364593

Mathematical Reviews number (MathSciNet)
MR3396997

Zentralblatt MATH identifier
1337.46020

Subjects
Primary: 46E10: Topological linear spaces of continuous, differentiable or analytic functions 46T05: Infinite-dimensional manifolds [See also 53Axx, 57N20, 58Bxx, 58Dxx] 46T10: Manifolds of mappings

Keywords
Ultradifferentiable functions convenient setting

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


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