| Title:
|
The $n$-dual space of the space of $p$-summable sequences (English) |
| Author:
|
Pangalela, Yosafat E. P. |
| Author:
|
Gunawan, Hendra |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
138 |
| Issue:
|
4 |
| Year:
|
2013 |
| Pages:
|
439-448 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In the theory of normed spaces, we have the concept of bounded linear functionals and dual spaces. Now, given an $n$-normed space, we are interested in bounded multilinear $n$-functionals and $n$-dual spaces. The concept of bounded multilinear $n$-functionals on an $n$-normed space was initially intoduced by White (1969), and studied further by Batkunde et al., and Gozali et al. (2010). In this paper, we revisit the definition of bounded multilinear $n$-functionals, introduce the concept of $n$-dual spaces, and then determine the $n$-dual spaces of $\ell^p$ spaces, when these spaces are not only equipped with the usual norm but also with some $n$-norms. (English) |
| Keyword:
|
$\ell^p$ space |
| Keyword:
|
$n$-normed space |
| Keyword:
|
$n$-dual space |
| MSC:
|
46B20 |
| MSC:
|
46B99 |
| MSC:
|
46C05 |
| MSC:
|
46C15 |
| MSC:
|
46C99 |
| idZBL:
|
Zbl 06260044 |
| idMR:
|
MR3231098 |
| DOI:
|
10.21136/MB.2013.143516 |
| . |
| Date available:
|
2013-11-09T20:28:09Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143516 |
| . |
| Reference:
|
[1] Batkunde, H., Gunawan, H., Pangalela, Y. E. P.: Bounded linear functionals on the $n$-normed space of $p$-summable sequences.(to appear) in Acta Univ. M. Belii Ser. Math. |
| Reference:
|
[2] Gähler, S.: Investigations on generalized $m$-metric spaces. I.German Math. Nachr. 40 (1969), 165-189. |
| Reference:
|
[3] Gähler, S.: Investigations on generalized $m$-metric spaces. II.German Math. Nachr. 40 (1969), 229-264. |
| Reference:
|
[4] Gähler, S.: Investigations on generalized $m$-metric spaces. III.German Math. Nachr. 41 (1969), 23-26. |
| Reference:
|
[5] Gozali, S. M., Gunawan, H., Neswan, O.: On $n$-norms and bounded $n$-linear functionals in a Hilbert space.Ann. Funct. Anal. AFA 1 (2010), 72-79, electronic only. Zbl 1208.46006, MR 2755461 |
| Reference:
|
[6] Gunawan, H.: The space of $p$-summable sequences and its natural $n$-norm.Bull. Aust. Math. Soc. 64 (2001), 137-147. Zbl 1002.46007, MR 1848086, 10.1017/S0004972700019754 |
| Reference:
|
[7] Gunawan, H., Setya-Budhi, W., Mashadi, M., Gemawati, S.: On volumes of $n$-dimensional parallelepipeds in $\ell^p$ spaces.Publ. Elektroteh. Fak., Univ. Beogr., Ser. Mat. 16 (2005), 48-54. MR 2164275 |
| Reference:
|
[8] Kreyszig, E.: Introductory Functional Analysis with Applications.Wiley Classics Library. John Wiley & Sons New York (1978). Zbl 0368.46014, MR 0467220 |
| Reference:
|
[9] Miličić, P. M.: On the Gram-Schmidt projection in normed spaces.Publ. Elektroteh. Fak., Univ. Beogr., Ser. Mat. 4 (1993), 89-96. Zbl 0819.46010, MR 1295606 |
| Reference:
|
[10] Pangalela, Y. E. P.: Representation of linear 2-functionals on space $\ell^p$.Indonesian Master Thesis, Institut Teknologi Bandung (2012). |
| Reference:
|
[11] White, A. G.: $2$-Banach spaces.Math. Nachr. 42 (1969), 43-60. Zbl 0185.20003, MR 0257716, 10.1002/mana.19690420104 |
| Reference:
|
[12] Wibawa-Kusumah, R. A., Gunawan, H.: Two equivalent $n$-norms on the space of $p$-summable sequences.Period. Math. Hung. 67 (2013), 63-69. MR 3090825, 10.1007/s10998-013-6129-4 |
| . |
