Strictly convex space : Strong orthogonality and Conjugate diameters
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arXiv
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| Formato: | Preprint |
| Publicado: |
2014
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| author | Sain, Debmalya Paul, Kallol Jha, Kanhaiya |
| author_facet | Sain, Debmalya Paul, Kallol Jha, Kanhaiya |
| contents | In a normed linear space X an element x is said to be orthogonal to another element y in the sense of Birkhoff-James, written as $ x \perp_{B}y, $ iff $ \| x \| \leq \| x + λy \| $ for all scalars $ λ.$ We prove that a normed linear space X is strictly convex iff for any two elements x, y of the unit sphere $ S_X$, $ x \perp_{B}y $ implies $ \| x + λy \| > 1~ \forall~ λ\neq 0. $ We apply this result to find a necessary and sufficient condition for a Hamel basis to be a strongly orthonormal Hamel basis in the sense of Birkhoff-James in a finite dimensional real strictly convex space X. Applying the result we give an estimation for lower bounds of $ \| tx+(1-t)y\|, t \in [0,1] $ and $ \| y + λx \|, ~\forall ~λ$ for all elements $ x,y \in S_X $ with $ x \perp_B y. $ We find a necessary and sufficient condition for the existence of conjugate diameters through the points $ e_1,e_2 \in ~S_X $ in a real strictly convex space of dimension 2. The concept of generalized conjuagte diameters is then developed for a real strictly convex smooth space of finite dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1411_1464 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | Strictly convex space : Strong orthogonality and Conjugate diameters Sain, Debmalya Paul, Kallol Jha, Kanhaiya Functional Analysis Primary 46B20, Secondary 47A30 In a normed linear space X an element x is said to be orthogonal to another element y in the sense of Birkhoff-James, written as $ x \perp_{B}y, $ iff $ \| x \| \leq \| x + λy \| $ for all scalars $ λ.$ We prove that a normed linear space X is strictly convex iff for any two elements x, y of the unit sphere $ S_X$, $ x \perp_{B}y $ implies $ \| x + λy \| > 1~ \forall~ λ\neq 0. $ We apply this result to find a necessary and sufficient condition for a Hamel basis to be a strongly orthonormal Hamel basis in the sense of Birkhoff-James in a finite dimensional real strictly convex space X. Applying the result we give an estimation for lower bounds of $ \| tx+(1-t)y\|, t \in [0,1] $ and $ \| y + λx \|, ~\forall ~λ$ for all elements $ x,y \in S_X $ with $ x \perp_B y. $ We find a necessary and sufficient condition for the existence of conjugate diameters through the points $ e_1,e_2 \in ~S_X $ in a real strictly convex space of dimension 2. The concept of generalized conjuagte diameters is then developed for a real strictly convex smooth space of finite dimension. |
| title | Strictly convex space : Strong orthogonality and Conjugate diameters |
| topic | Functional Analysis Primary 46B20, Secondary 47A30 |
| url | https://arxiv.org/abs/1411.1464 |