On the closure of the sum of closed subspaces
We give necessary and sufficient conditions for the sum of closed subspaces of a Hilbert space to be closed. Specifically, we show that the sum will be closed if and only if the angle between the subspaces is not zero, or if and only if the projection of either space into the orthogonal complement...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2001-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171201005324 |
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| _version_ | 1832559452939616256 |
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| author | Irwin E. Schochetman Robert L. Smith Sze-Kai Tsui |
| author_facet | Irwin E. Schochetman Robert L. Smith Sze-Kai Tsui |
| author_sort | Irwin E. Schochetman |
| collection | DOAJ |
| description | We give necessary and sufficient conditions for the sum of closed subspaces of a Hilbert space to be closed. Specifically, we show that the sum will be closed if and only if the angle between the subspaces is not zero, or if and only if the projection of either space into
the orthogonal complement of the other is closed. We also give sufficient conditions for the sum to be closed in terms of the relevant orthogonal projections. As a consequence, we obtain sufficient conditions for the existence of an optimal solution to an abstract quadratic programming problem in terms of the kernels of the cost and constraint operators. |
| format | Article |
| id | doaj-art-dc45672dc3264a50891151d8940ede3d |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2001-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-dc45672dc3264a50891151d8940ede3d2025-02-03T01:30:09ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252001-01-0126525726710.1155/S0161171201005324On the closure of the sum of closed subspacesIrwin E. Schochetman0Robert L. Smith1Sze-Kai Tsui2Mathematics and Statistics, Oakland University, Rochester, MI 48309, USAIndustrial and Operations Engineering, The University of Michigan, Ann Arbor, MI 48109, USAMathematics and Statistics, Oakland University, Rochester, MI 48309, USAWe give necessary and sufficient conditions for the sum of closed subspaces of a Hilbert space to be closed. Specifically, we show that the sum will be closed if and only if the angle between the subspaces is not zero, or if and only if the projection of either space into the orthogonal complement of the other is closed. We also give sufficient conditions for the sum to be closed in terms of the relevant orthogonal projections. As a consequence, we obtain sufficient conditions for the existence of an optimal solution to an abstract quadratic programming problem in terms of the kernels of the cost and constraint operators.http://dx.doi.org/10.1155/S0161171201005324 |
| spellingShingle | Irwin E. Schochetman Robert L. Smith Sze-Kai Tsui On the closure of the sum of closed subspaces International Journal of Mathematics and Mathematical Sciences |
| title | On the closure of the sum of closed subspaces |
| title_full | On the closure of the sum of closed subspaces |
| title_fullStr | On the closure of the sum of closed subspaces |
| title_full_unstemmed | On the closure of the sum of closed subspaces |
| title_short | On the closure of the sum of closed subspaces |
| title_sort | on the closure of the sum of closed subspaces |
| url | http://dx.doi.org/10.1155/S0161171201005324 |
| work_keys_str_mv | AT irwineschochetman ontheclosureofthesumofclosedsubspaces AT robertlsmith ontheclosureofthesumofclosedsubspaces AT szekaitsui ontheclosureofthesumofclosedsubspaces |