A New Semi-Inner Product and <i>p<sub>n</sub></i>-Angle in the Space of <i>p</i>-Summable Sequences
In this paper, we propose a definition for a semi-inner product in the space of <i>p</i>-summable sequences equipped with an <i>n</i>-norm. Using this definition, we introduce the concepts of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML"...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-07-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/14/3139 |
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| author | Muh Nur Mawardi Bahri Anna Islamiyati Harmanus Batkunde |
| author_facet | Muh Nur Mawardi Bahri Anna Islamiyati Harmanus Batkunde |
| author_sort | Muh Nur |
| collection | DOAJ |
| description | In this paper, we propose a definition for a semi-inner product in the space of <i>p</i>-summable sequences equipped with an <i>n</i>-norm. Using this definition, we introduce the concepts of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>p</mi><mi>n</mi></msub></semantics></math></inline-formula>-orthogonality and the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>p</mi><mi>n</mi></msub></semantics></math></inline-formula>-angle between two vectors in the space of <i>p</i>-summable sequences. For the special case <i>n</i> = 1, these concepts are identical to previous studies. We also introduce the notion of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>p</mi><mi>n</mi></msub></semantics></math></inline-formula>-angle between one-dimensional subspaces and arbitrary-dimensional subspaces. The authors believe that the results obtained in this paper are very significant, especially in the theory of <i>n</i>-normed space in functional analysis. |
| first_indexed | 2024-03-11T00:51:59Z |
| format | Article |
| id | doaj.art-a45271399b174cd087f7c7fad579808c |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T00:51:59Z |
| publishDate | 2023-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-a45271399b174cd087f7c7fad579808c2023-11-18T20:21:07ZengMDPI AGMathematics2227-73902023-07-011114313910.3390/math11143139A New Semi-Inner Product and <i>p<sub>n</sub></i>-Angle in the Space of <i>p</i>-Summable SequencesMuh Nur0Mawardi Bahri1Anna Islamiyati2Harmanus Batkunde3Department of Mathematics, Hasanuddin University, Makassar 90245, IndonesiaDepartment of Mathematics, Hasanuddin University, Makassar 90245, IndonesiaDepartment of Statistics, Hasanuddin University, Makassar 90245, IndonesiaDepartment of Mathematics, Pattimura University, Ambon 97233, IndonesiaIn this paper, we propose a definition for a semi-inner product in the space of <i>p</i>-summable sequences equipped with an <i>n</i>-norm. Using this definition, we introduce the concepts of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>p</mi><mi>n</mi></msub></semantics></math></inline-formula>-orthogonality and the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>p</mi><mi>n</mi></msub></semantics></math></inline-formula>-angle between two vectors in the space of <i>p</i>-summable sequences. For the special case <i>n</i> = 1, these concepts are identical to previous studies. We also introduce the notion of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>p</mi><mi>n</mi></msub></semantics></math></inline-formula>-angle between one-dimensional subspaces and arbitrary-dimensional subspaces. The authors believe that the results obtained in this paper are very significant, especially in the theory of <i>n</i>-normed space in functional analysis.https://www.mdpi.com/2227-7390/11/14/3139semi-inner product<i>p<sub>n</sub></i>-orthogonality<i>p<sub>n</sub></i>-angle<i>n</i>-norm |
| spellingShingle | Muh Nur Mawardi Bahri Anna Islamiyati Harmanus Batkunde A New Semi-Inner Product and <i>p<sub>n</sub></i>-Angle in the Space of <i>p</i>-Summable Sequences Mathematics semi-inner product <i>p<sub>n</sub></i>-orthogonality <i>p<sub>n</sub></i>-angle <i>n</i>-norm |
| title | A New Semi-Inner Product and <i>p<sub>n</sub></i>-Angle in the Space of <i>p</i>-Summable Sequences |
| title_full | A New Semi-Inner Product and <i>p<sub>n</sub></i>-Angle in the Space of <i>p</i>-Summable Sequences |
| title_fullStr | A New Semi-Inner Product and <i>p<sub>n</sub></i>-Angle in the Space of <i>p</i>-Summable Sequences |
| title_full_unstemmed | A New Semi-Inner Product and <i>p<sub>n</sub></i>-Angle in the Space of <i>p</i>-Summable Sequences |
| title_short | A New Semi-Inner Product and <i>p<sub>n</sub></i>-Angle in the Space of <i>p</i>-Summable Sequences |
| title_sort | new semi inner product and i p sub n sub i angle in the space of i p i summable sequences |
| topic | semi-inner product <i>p<sub>n</sub></i>-orthogonality <i>p<sub>n</sub></i>-angle <i>n</i>-norm |
| url | https://www.mdpi.com/2227-7390/11/14/3139 |
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