Linear algebra and differential geometry on abstract Hilbert space
Isomorphisms of separable Hilbert spaces are analogous to isomorphisms of n-dimensional vector spaces. However, while n-dimensional spaces in applications are always realized as the Euclidean space Rn, Hilbert spaces admit various useful realizations as spaces of functions. In the paper this simple...
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| Format: | Article |
| Language: | English |
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Wiley
2005-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/IJMMS.2005.2241 |
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| _version_ | 1832554137965821952 |
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| author | Alexey A. Kryukov |
| author_facet | Alexey A. Kryukov |
| author_sort | Alexey A. Kryukov |
| collection | DOAJ |
| description | Isomorphisms of separable Hilbert spaces are analogous to
isomorphisms of n-dimensional vector spaces. However,
while n-dimensional spaces in applications are always realized
as the Euclidean space Rn, Hilbert spaces admit various useful
realizations as spaces of functions. In the paper this simple
observation is used to construct a fruitful formalism of local
coordinates on Hilbert manifolds. Images of charts on manifolds in
the formalism are allowed to belong to arbitrary Hilbert spaces of
functions including spaces of generalized functions. Tensor
equations then describe families of functional equations on
various spaces of functions. The formalism itself and its
applications in linear algebra, differential equations, and
differential geometry are carefully analyzed. |
| format | Article |
| id | doaj-art-22634d99426946cdb7b1ffde5e53f5f1 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2005-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-22634d99426946cdb7b1ffde5e53f5f12025-02-03T05:52:16ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252005-01-012005142241227510.1155/IJMMS.2005.2241Linear algebra and differential geometry on abstract Hilbert spaceAlexey A. Kryukov0Department of Mathematics, University of Wisconsin Colleges, 780 Regent Street, Madison 53708, WI, USAIsomorphisms of separable Hilbert spaces are analogous to isomorphisms of n-dimensional vector spaces. However, while n-dimensional spaces in applications are always realized as the Euclidean space Rn, Hilbert spaces admit various useful realizations as spaces of functions. In the paper this simple observation is used to construct a fruitful formalism of local coordinates on Hilbert manifolds. Images of charts on manifolds in the formalism are allowed to belong to arbitrary Hilbert spaces of functions including spaces of generalized functions. Tensor equations then describe families of functional equations on various spaces of functions. The formalism itself and its applications in linear algebra, differential equations, and differential geometry are carefully analyzed.http://dx.doi.org/10.1155/IJMMS.2005.2241 |
| spellingShingle | Alexey A. Kryukov Linear algebra and differential geometry on abstract Hilbert space International Journal of Mathematics and Mathematical Sciences |
| title | Linear algebra and differential geometry on abstract Hilbert space |
| title_full | Linear algebra and differential geometry on abstract Hilbert space |
| title_fullStr | Linear algebra and differential geometry on abstract Hilbert space |
| title_full_unstemmed | Linear algebra and differential geometry on abstract Hilbert space |
| title_short | Linear algebra and differential geometry on abstract Hilbert space |
| title_sort | linear algebra and differential geometry on abstract hilbert space |
| url | http://dx.doi.org/10.1155/IJMMS.2005.2241 |
| work_keys_str_mv | AT alexeyakryukov linearalgebraanddifferentialgeometryonabstracthilbertspace |