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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Finitely generated submodules of differentiable functions


Author: B. Roth
Journal: Proc. Amer. Math. Soc. 34 (1972), 433-439
MSC: Primary 46E15
DOI: https://doi.org/10.1090/S0002-9939-1972-0300072-1
MathSciNet review: 0300072
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Abstract: Suppose $ {[{\mathcal{E}^m}(\Omega )]^p}$ is the Cartesian product of the space of real-valued m-times continuously differentiable functions on an open set $ \Omega $ in $ {R^n}$ with itself p-times where m is finite and $ \Omega $ is connected. $ {[{\mathcal{E}^m}(\Omega )]^p}$ is a $ {\mathcal{E}^m}(\Omega )$-module. The finitely generated submodules of $ {[{\mathcal{E}^m}(\Omega )]^p}$ are $ {\text{im}}(F)$ where $ F:{[{\mathcal{E}^m}(\Omega )]^q} \to {[{\mathcal{E}^m}(\Omega )]^p}$ is a $ p \times q$ matrix $ {({f_{ij}})_{1 \leqq i \leqq p,}}_{1 \leqq j \leqq q},{f_{ij}} \in {\mathcal{E}^m}(\Omega )$. In the present paper, it is shown that $ {\text{im}}(F)$ is closed in $ {[{\mathcal{E}^m}(\Omega )]^p}$ if and only if the rank of the matrix $ {({f_{ij}}(x))_{1 \leqq i \leqq p,1 \leqq j \leqq q}}$ is constant for $ x \in \Omega $. Applications are made to systems of division problems for distributions.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0300072-1
Keywords: Spaces of differentiable functions, modules of differentiable functions, finitely generated submodules, spaces of distributions, division of distributions
Article copyright: © Copyright 1972 American Mathematical Society