Inner Product Fuzzy Quasilinear Spaces and Some Fuzzy Sequence Spaces
It has been shown that the class of fuzzy sets has a quasilinear space structure. In addition, various norms are defined on this class, and it is given that the class of fuzzy sets is a normed quasilinear space with these norms. In this study, we first developed the algebraic structure of the class...
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| Format: | Article |
| Language: | English |
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Wiley
2022-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2022/2466817 |
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| author | Yılmaz Yılmaz Hacer Bozkurt Halise Levent Ümit Çetinkaya |
| author_facet | Yılmaz Yılmaz Hacer Bozkurt Halise Levent Ümit Çetinkaya |
| author_sort | Yılmaz Yılmaz |
| collection | DOAJ |
| description | It has been shown that the class of fuzzy sets has a quasilinear space structure. In addition, various norms are defined on this class, and it is given that the class of fuzzy sets is a normed quasilinear space with these norms. In this study, we first developed the algebraic structure of the class of fuzzy sets Fℝn and gave definitions such as quasilinear independence, dimension, and the algebraic basis in these spaces. Then, with special norms, namely, uq=∫01supx∈uαxqdα1/q where 1≤q≤∞, we stated that Fℝn,uq is a complete normed space. Furthermore, we introduced an inner product in this space for the case q=2. The inner product must be in the formu,v=∫01uα,vα Kℝndα=∫01a,bℝndα :a∈uα,b∈vα. For u,v∈Fℝn. We also proved that the parallelogram law can only be provided in the regular subspace, not in the entire of Fℝn. Finally, we showed that a special class of fuzzy number sequences is a Hilbert quasilinear space. |
| format | Article |
| id | doaj-art-6591e3b8f3ea4ac5a249f8339c7f2403 |
| institution | Kabale University |
| issn | 2314-4785 |
| language | English |
| publishDate | 2022-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-6591e3b8f3ea4ac5a249f8339c7f24032025-02-03T05:49:25ZengWileyJournal of Mathematics2314-47852022-01-01202210.1155/2022/2466817Inner Product Fuzzy Quasilinear Spaces and Some Fuzzy Sequence SpacesYılmaz Yılmaz0Hacer Bozkurt1Halise Levent2Ümit Çetinkaya3Department of MathematicsDepartment of MathematicsDepartment of MathematicsDepartment of MathematicsIt has been shown that the class of fuzzy sets has a quasilinear space structure. In addition, various norms are defined on this class, and it is given that the class of fuzzy sets is a normed quasilinear space with these norms. In this study, we first developed the algebraic structure of the class of fuzzy sets Fℝn and gave definitions such as quasilinear independence, dimension, and the algebraic basis in these spaces. Then, with special norms, namely, uq=∫01supx∈uαxqdα1/q where 1≤q≤∞, we stated that Fℝn,uq is a complete normed space. Furthermore, we introduced an inner product in this space for the case q=2. The inner product must be in the formu,v=∫01uα,vα Kℝndα=∫01a,bℝndα :a∈uα,b∈vα. For u,v∈Fℝn. We also proved that the parallelogram law can only be provided in the regular subspace, not in the entire of Fℝn. Finally, we showed that a special class of fuzzy number sequences is a Hilbert quasilinear space.http://dx.doi.org/10.1155/2022/2466817 |
| spellingShingle | Yılmaz Yılmaz Hacer Bozkurt Halise Levent Ümit Çetinkaya Inner Product Fuzzy Quasilinear Spaces and Some Fuzzy Sequence Spaces Journal of Mathematics |
| title | Inner Product Fuzzy Quasilinear Spaces and Some Fuzzy Sequence Spaces |
| title_full | Inner Product Fuzzy Quasilinear Spaces and Some Fuzzy Sequence Spaces |
| title_fullStr | Inner Product Fuzzy Quasilinear Spaces and Some Fuzzy Sequence Spaces |
| title_full_unstemmed | Inner Product Fuzzy Quasilinear Spaces and Some Fuzzy Sequence Spaces |
| title_short | Inner Product Fuzzy Quasilinear Spaces and Some Fuzzy Sequence Spaces |
| title_sort | inner product fuzzy quasilinear spaces and some fuzzy sequence spaces |
| url | http://dx.doi.org/10.1155/2022/2466817 |
| work_keys_str_mv | AT yılmazyılmaz innerproductfuzzyquasilinearspacesandsomefuzzysequencespaces AT hacerbozkurt innerproductfuzzyquasilinearspacesandsomefuzzysequencespaces AT haliselevent innerproductfuzzyquasilinearspacesandsomefuzzysequencespaces AT umitcetinkaya innerproductfuzzyquasilinearspacesandsomefuzzysequencespaces |