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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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
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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| Summary: | 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. |
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| ISSN: | 0161-1712 1687-0425 |