Relationship between Generalized Orthogonality and Gâteaux Derivative
This paper investigates the relationship between generalized orthogonality and Gâteaux derivative of the norm in a normed linear space. It is shown that the Gâteaux derivative of <i>x</i> in the <i>y</i> direction is zero when the norm is Gâteaux differentiable in the <i&g...
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MDPI AG
2024-01-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/12/3/364 |
| _version_ | 1797318483440041984 |
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| author | Peixuan Xu Donghai Ji Hongxu Zhang |
| author_facet | Peixuan Xu Donghai Ji Hongxu Zhang |
| author_sort | Peixuan Xu |
| collection | DOAJ |
| description | This paper investigates the relationship between generalized orthogonality and Gâteaux derivative of the norm in a normed linear space. It is shown that the Gâteaux derivative of <i>x</i> in the <i>y</i> direction is zero when the norm is Gâteaux differentiable in the <i>y</i> direction at <i>x</i> and <i>x</i> and <i>y</i> satisfy certain generalized orthogonality conditions. A case where <i>x</i> and <i>y</i> are approximately orthogonal is also analyzed and the value range of the Gâteaux derivative in this case is given. Moreover, two concepts are introduced: the angle between vectors in normed linear space and the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>⊥</mo><mo>Δ</mo></msub></semantics></math></inline-formula> coordinate system in a smooth Minkowski plane. Relevant examples are given at the end of the paper. |
| first_indexed | 2024-03-08T03:53:03Z |
| format | Article |
| id | doaj.art-c95f9b5a9b774be0b596e1dd8a49d16b |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-08T03:53:03Z |
| publishDate | 2024-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-c95f9b5a9b774be0b596e1dd8a49d16b2024-02-09T15:18:05ZengMDPI AGMathematics2227-73902024-01-0112336410.3390/math12030364Relationship between Generalized Orthogonality and Gâteaux DerivativePeixuan Xu0Donghai Ji1Hongxu Zhang2Department of Applied Mathematics, Harbin University of Science and Technology, Harbin 150080, ChinaDepartment of Applied Mathematics, Harbin University of Science and Technology, Harbin 150080, ChinaDepartment of Applied Mathematics, Harbin University of Science and Technology, Harbin 150080, ChinaThis paper investigates the relationship between generalized orthogonality and Gâteaux derivative of the norm in a normed linear space. It is shown that the Gâteaux derivative of <i>x</i> in the <i>y</i> direction is zero when the norm is Gâteaux differentiable in the <i>y</i> direction at <i>x</i> and <i>x</i> and <i>y</i> satisfy certain generalized orthogonality conditions. A case where <i>x</i> and <i>y</i> are approximately orthogonal is also analyzed and the value range of the Gâteaux derivative in this case is given. Moreover, two concepts are introduced: the angle between vectors in normed linear space and the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>⊥</mo><mo>Δ</mo></msub></semantics></math></inline-formula> coordinate system in a smooth Minkowski plane. Relevant examples are given at the end of the paper.https://www.mdpi.com/2227-7390/12/3/364Gâteaux derivativeisosceles orthogonalityBirkhoff orthogonalityapproximate orthogonality |
| spellingShingle | Peixuan Xu Donghai Ji Hongxu Zhang Relationship between Generalized Orthogonality and Gâteaux Derivative Mathematics Gâteaux derivative isosceles orthogonality Birkhoff orthogonality approximate orthogonality |
| title | Relationship between Generalized Orthogonality and Gâteaux Derivative |
| title_full | Relationship between Generalized Orthogonality and Gâteaux Derivative |
| title_fullStr | Relationship between Generalized Orthogonality and Gâteaux Derivative |
| title_full_unstemmed | Relationship between Generalized Orthogonality and Gâteaux Derivative |
| title_short | Relationship between Generalized Orthogonality and Gâteaux Derivative |
| title_sort | relationship between generalized orthogonality and gateaux derivative |
| topic | Gâteaux derivative isosceles orthogonality Birkhoff orthogonality approximate orthogonality |
| url | https://www.mdpi.com/2227-7390/12/3/364 |
| work_keys_str_mv | AT peixuanxu relationshipbetweengeneralizedorthogonalityandgateauxderivative AT donghaiji relationshipbetweengeneralizedorthogonalityandgateauxderivative AT hongxuzhang relationshipbetweengeneralizedorthogonalityandgateauxderivative |