Absence of Non-Trivial Fuzzy Inner Product Spaces and the Cauchy–Schwartz Inequality
First, we show that the non-trivial fuzzy inner product space under the linearity condition does not exist, which means a fuzzy inner product space with linearity produces only a crisp real number for each pair of vectors. If the positive-definiteness is added to the condition, then the Cauchy–Schwa...
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MDPI AG
2020-04-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/4/571 |
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doaj-60149ffb6a584b53b7187fba8c7985a12020-11-25T02:44:06ZengMDPI AGMathematics2227-73902020-04-01857157110.3390/math8040571Absence of Non-Trivial Fuzzy Inner Product Spaces and the Cauchy–Schwartz InequalityTaechang Byun0Ji Eun Lee1Keun Young Lee2Jin Hee Yoon3Faculty of Mathematics and Statistics, Sejong University, Seoul 05006, KoreaFaculty of Mathematics and Statistics, Sejong University, Seoul 05006, KoreaFaculty of Mathematics and Statistics, Sejong University, Seoul 05006, KoreaFaculty of Mathematics and Statistics, Sejong University, Seoul 05006, KoreaFirst, we show that the non-trivial fuzzy inner product space under the linearity condition does not exist, which means a fuzzy inner product space with linearity produces only a crisp real number for each pair of vectors. If the positive-definiteness is added to the condition, then the Cauchy–Schwartz inequality is also proved.https://www.mdpi.com/2227-7390/8/4/571fuzzy inner product spaceCauchy–Schwartz inequalitylinearitypositive-definiteness |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
Taechang Byun Ji Eun Lee Keun Young Lee Jin Hee Yoon |
| spellingShingle |
Taechang Byun Ji Eun Lee Keun Young Lee Jin Hee Yoon Absence of Non-Trivial Fuzzy Inner Product Spaces and the Cauchy–Schwartz Inequality Mathematics fuzzy inner product space Cauchy–Schwartz inequality linearity positive-definiteness |
| author_facet |
Taechang Byun Ji Eun Lee Keun Young Lee Jin Hee Yoon |
| author_sort |
Taechang Byun |
| title |
Absence of Non-Trivial Fuzzy Inner Product Spaces and the Cauchy–Schwartz Inequality |
| title_short |
Absence of Non-Trivial Fuzzy Inner Product Spaces and the Cauchy–Schwartz Inequality |
| title_full |
Absence of Non-Trivial Fuzzy Inner Product Spaces and the Cauchy–Schwartz Inequality |
| title_fullStr |
Absence of Non-Trivial Fuzzy Inner Product Spaces and the Cauchy–Schwartz Inequality |
| title_full_unstemmed |
Absence of Non-Trivial Fuzzy Inner Product Spaces and the Cauchy–Schwartz Inequality |
| title_sort |
absence of non-trivial fuzzy inner product spaces and the cauchy–schwartz inequality |
| publisher |
MDPI AG |
| series |
Mathematics |
| issn |
2227-7390 |
| publishDate |
2020-04-01 |
| description |
First, we show that the non-trivial fuzzy inner product space under the linearity condition does not exist, which means a fuzzy inner product space with linearity produces only a crisp real number for each pair of vectors. If the positive-definiteness is added to the condition, then the Cauchy–Schwartz inequality is also proved. |
| topic |
fuzzy inner product space Cauchy–Schwartz inequality linearity positive-definiteness |
| url |
https://www.mdpi.com/2227-7390/8/4/571 |
| work_keys_str_mv |
AT taechangbyun absenceofnontrivialfuzzyinnerproductspacesandthecauchyschwartzinequality AT jieunlee absenceofnontrivialfuzzyinnerproductspacesandthecauchyschwartzinequality AT keunyounglee absenceofnontrivialfuzzyinnerproductspacesandthecauchyschwartzinequality AT jinheeyoon absenceofnontrivialfuzzyinnerproductspacesandthecauchyschwartzinequality |
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1724767387821015040 |