Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
$n$-normed space; $n$-Banach space; $b$-linear functional; Banach algebra; complex homeomorphism
$n$-normed space; $n$-Banach space; $b$-linear functional; Banach algebra; complex homeomorphism
Summary:
The idea of best approximation in linear $n$-normed space is presented and some examples showing various possibilities of best approximations in linear $n$-normed space is given. Also, we study strictly convex $n$-norm and enquire about the uniqueness of best approximations in $n$-normed linear space. Furthermore, best approximations in $n$-Hilbert space is discussed. Moreover, the notion of a Banach algebra in $n$-Banach space is presented and some examples are discussed. A set-theoretic property of invertible and noninvertible elements in an $n$-Banach algebra is explained and then topological divisor of zero in $n$-Banach algebra is defined. Finally, we introduce the notion of a complex homeomorphism in an $n$-Banach algebra and derive Gleason, Kahane, Zelazko type theorem with the help of complex $b$-homeomorphism in the case of $n$-Banach algebra.
References:
[1] Diminnie, C., Gähler, S., White, A.: 2-inner product spaces. Demonstr. Math. 6 (1973), 525-536. DOI 10.1515/dema-1977-0115 | MR 0365099 | Zbl 0296.46022
[2] Freese, R. W., Cho, Y. J.: Geometry of Linear 2-Normed Spaces. Nova Science Publishers, New York (2001). MR 2127507 | Zbl 1051.46001
[3] Gähler, S.: Lineare 2-normierte Räume. Math. Nachr. 28 (1964), 1-43 German. DOI 10.1002/mana.19640280102 | MR 0169021 | Zbl 0142.39803
[4] Gelfand, I. M.: Normierte Ringe. Rec. Math. Moscou, n. Ser. 9 (1941), 3-24 German. MR 0004726 | Zbl 0024.32002
[5] Gelfand, I. M., Neumark, M. A.: On the imbedding of normed rings into the ring of operators in Hilbert space. Mat. Sb., N. Ser. 12 (1943), 197-213. MR 0009426 | Zbl 0060.27006
[6] Ghosh, P., Samanta, T. K.: A few fixed point theorems in linear $n$-normed space. Available at https://arxiv.org/abs/2210.07849 (2022), 14 pages. DOI 10.48550/arXiv.2210.07849
[7] Ghosh, P., Samanta, T. K.: Representation of uniform boundedness principle and Hahn-Banach theorem in linear $n$-normed space. J. Anal. 30 (2022), 597-619. DOI 10.1007/s41478-021-00358-x | MR 4431102 | Zbl 1503.46019
[8] Ghosh, P., Samanta, T. K.: Slow convergence of sequence of $b$-linear functionals in linear $n$-normed space. Palest. J. Math. 12 (2023), 501-511. DOI 10.46698/o3961-3328-9819-i | MR 4582358 | Zbl 07920936
[9] Ghosh, P., Samanta, T. K.: Reflexivity of linear $n$-normed space with respect to $b$-linear functional. Facta Univ., Ser. Math. Inf. 39 (2024), 183-199. DOI 10.22190/FUMI201023013G | MR 4753218 | Zbl 1549.46003
[10] Guediri, H., Garayev, M. T., Sadraoui, H.: The Bergman space as a Banach algebra. New York J. Math. 21 (2015), 339-350. MR 3358547 | Zbl 1329.47034
[11] Gunawan, H., Mashadi, M.: On $n$-normed spaces. Int. J. Math. Math. Sci. 27 (2001), 631-639. DOI 10.1155/S0161171201010675 | MR 1873126 | Zbl 1006.46006
[12] Misiak, A.: $n$-inner product spaces. Math. Nachr. 140 (1989), 299-319. DOI 10.1002/mana.19891400121 | MR 1015402 | Zbl 0673.46012
[13] Mohammad, N., Siddiqi, A. H.: On 2-Banach algebras. Aligarh Bull. Math. 12 (1989), 51-60. Zbl 0751.46030
[14] Nagumo, M.: Einige analytische Untersuchungen in linearen, metrischen Eingen. Japanese J. Math. 13 (1936), 61-80 German \99999JFM99999 62.0449.01. DOI 10.4099/jjm1924.13.0_61
[15] Rudin, W.: Functional Analysis. McGraw-Hill Series in Higher Mathematics. Mc Graw Hill, New York (1973). MR 0365062 | Zbl 0253.46001
[16] Soenjaya, A. L.: The open mapping theorem in $n$-Banach space. Int. J. Pure Appl. Math. 76 (2012), 593-598. Zbl 1284.46005
[17] Srivastava, N., Bhattacharya, S., Lal, S. N.: 2-normed algebras. I. Publ. Inst. Math., Nouv. Sér. 88 (2010), 111-121. DOI 10.2298/PIM1002111S | MR 2762625 | Zbl 1263.46008
[18] goglu, R. Tapdı: On the Banach algebra structure for $C^{(n)}$ of the bidisc and related topics. Ill. J. Math. 64 (2020), 185-197. DOI 10.1215/00192082-8303477 | MR 4092955 | Zbl 1455.46055
[19] Tapdigoğlu, R., Kösem, Ü.: Some results on 2-Banach algebras. Electron. J. Math. Anal. Appl. 4 (2016), 11-14. MR 3396126 | Zbl 1463.46079
[20] A. G. White, Jr.: 2-Banach spaces. Math. Nachr. 42 (1969), 43-60. DOI 10.1002/mana.19690420104 | MR 0257716 | Zbl 0185.20003
Search
Partner of
