Abstract
Distributional tensor fields can be regarded as multilinear mappings on smooth tensor fields with distributional values or as (classical) tensor fields with distributional coefficients. We show that the corresponding isomorphisms hold also in the bornological setting.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Blyth, T.S.: Module Theory. Clarendon Press, Oxford (1997)
Bourbaki, N.: Algebra I, Chapt. 1–3. Elements of Mathematics. Springer, Berlin (1970)
Brickell, F., Clark, R.S.: Differentiable Manifolds: An Introduction. Van Nostrand Reinhold Company, London (1970)
Capelle, J.: Convolution on homogeneous spaces. PhD thesis, University of Groningen (1996)
Greub, W., Halperin, S., Vanstone, R.: Connections, curvature, and cohomology. Vol. I: De Rham cohomology of manifolds and vector bundles, volume 47 of Pure and Applied Mathematics. Academic Press, New York (1972)
Grosser, M., Kunzinger, M., Oberguggenberger, M., Steinbauer, R.: Geometric Theory of Generalized Functions with Applications to General Relativity. Kluwer Academic Publishers, Dordrecht (2001)
Grosser, M., Kunzinger, M., Steinbauer, R., Vickers, J.A.: A global theory of algebras of generalized functions. II. Tensor distributions. N. Y. J. Math. 18, 139–199 (2012)
Hörmander, L.: The analysis of linear partial differential operators. I. In: Classics in Mathematics. Springer, Berlin (2003)
Jarchow, H.: Locally Convex Spaces. B. G.Teubner, Stuttgart (1981)
Kriegl, A., Michor, P.W.: The Convenient Setting of Global Analysis, Mathematical Surveys and Monographs, vol. 53. American Mathematical Society, Providence (1997)
Kucera, J., McKennon, K.: Continuity of multiplication of distributions. Int. J. Math. Math. Sci 4, 819–822 (1981)
Lang, S.: Fundamentals of Differential Geometry. Springer, New York (1999)
Navarro González, J.A., Sancho de Salas, J.B.: \(C^\infty \)-Differentiable Spaces. Springer, Berlin (2003)
Schaefer, H.H.: Topological Vector Spaces. Springer, New York (1971)
Treves, F.: Topological Vector Spaces, Distributions and Kernels. Academic Press, New York (1976)
Acknowledgments
This research has been supported by START-project Y237 and project P20525 of the Austrian Science Fund and the Doctoral College ’Differential Geometry and Lie Groups’ of the University of Vienna.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Cap.
Rights and permissions
About this article
Cite this article
Nigsch, E.A. Bornologically isomorphic representations of distributions on manifolds. Monatsh Math 170, 49–63 (2013). https://doi.org/10.1007/s00605-012-0442-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-012-0442-5