Orthogonally Complemented Subspaces in Banach Spaces |
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| Authors: | Henryk Hudzik Yuwen Wang Ruli Sha |
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| Institution: | 1. Faculty of Mathematics and Computer Science , Adam Mickiewicz University , Umultowska, Poznań, Poland hudzik@amu.edu.pl;3. School of Mathematics and Computer Science, Harbin Normal University , Harbin, People's Republic of China;4. Department of Mathematics , Hulunbeir College , Hailar, People's Republic of China |
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| Abstract: | In this paper, we extend the Moreau (Riesz) decomposition theorem from Hilbert spaces to Banach spaces. Criteria for a closed subspace to be (strongly) orthogonally complemented in a Banach space are given. We prove that every closed subspace of a Banach space X with dim X ≥ 3 (dim X ≤ 2) is strongly orthognally complemented if and only if the Banach space X is isometric to a Hilbert space (resp. strictly convex), which is complementary to the well-known result saying that every closed subspace of a Banach space X is topologically complemented if and only if the Banach space X is isomorphic to a Hilbert space. |
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| Keywords: | Banach space Duality mapping Hilbert space Orthogonally complemented subspace Strict convexity |
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