Norm Attaining Multilinear Forms on 𝐿1(𝝁)
Given an arbitrary measure 𝜇, this study shows that the set of norm attaining multilinear forms is not dense in the space of all continuous multilinear forms on 𝐿1(𝜇). However, we have the density if and only if 𝜇 is purely atomic. Furthermore, the study presents an example of a Banach space 𝑋 in wh...
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| Format: | Article |
| Language: | English |
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Wiley
2008-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2008/328481 |
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| _version_ | 1832561990011191296 |
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| author | Yousef Saleh |
| author_facet | Yousef Saleh |
| author_sort | Yousef Saleh |
| collection | DOAJ |
| description | Given an arbitrary measure 𝜇, this study shows that the set of norm attaining multilinear
forms is not dense in the space of all continuous multilinear forms on 𝐿1(𝜇). However, we have the density if and only if 𝜇 is purely atomic. Furthermore, the study presents an example of a
Banach space 𝑋 in which the set of norm attaining operators from 𝑋 into 𝑋∗ is dense in the
space of all bounded linear operators 𝐿(𝑋,𝑋∗). In contrast, the set of norm attaining bilinear
forms on 𝑋 is not dense in the space of continuous bilinear forms on 𝑋. |
| format | Article |
| id | doaj-art-8d48f499a0284461af3cb680758da3f8 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2008-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-8d48f499a0284461af3cb680758da3f82025-02-03T01:23:47ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252008-01-01200810.1155/2008/328481328481Norm Attaining Multilinear Forms on 𝐿1(𝝁)Yousef Saleh0Mathematics Department, Hebron University, P.O. Box 40, Hebron, West Bank, PalestineGiven an arbitrary measure 𝜇, this study shows that the set of norm attaining multilinear forms is not dense in the space of all continuous multilinear forms on 𝐿1(𝜇). However, we have the density if and only if 𝜇 is purely atomic. Furthermore, the study presents an example of a Banach space 𝑋 in which the set of norm attaining operators from 𝑋 into 𝑋∗ is dense in the space of all bounded linear operators 𝐿(𝑋,𝑋∗). In contrast, the set of norm attaining bilinear forms on 𝑋 is not dense in the space of continuous bilinear forms on 𝑋.http://dx.doi.org/10.1155/2008/328481 |
| spellingShingle | Yousef Saleh Norm Attaining Multilinear Forms on 𝐿1(𝝁) International Journal of Mathematics and Mathematical Sciences |
| title | Norm Attaining Multilinear Forms on 𝐿1(𝝁) |
| title_full | Norm Attaining Multilinear Forms on 𝐿1(𝝁) |
| title_fullStr | Norm Attaining Multilinear Forms on 𝐿1(𝝁) |
| title_full_unstemmed | Norm Attaining Multilinear Forms on 𝐿1(𝝁) |
| title_short | Norm Attaining Multilinear Forms on 𝐿1(𝝁) |
| title_sort | norm attaining multilinear forms on 𝐿1 𝝁 |
| url | http://dx.doi.org/10.1155/2008/328481 |
| work_keys_str_mv | AT yousefsaleh normattainingmultilinearformsonl1μ |