| Title:
|
On some geometric properties of certain Köthe sequence spaces (English) |
| Author:
|
Cui, Yunan |
| Author:
|
Hudzik, Henryk |
| Author:
|
Zhang, Tao |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
124 |
| Issue:
|
2 |
| Year:
|
1999 |
| Pages:
|
303-314 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
It is proved that if a Kothe sequence space $X$ is monotone complete and has the weakly convergent sequence coefficient WCS$(X)>1$, then $X$ is order continuous. It is shown that a weakly sequentially complete Kothe sequence space $X$ is compactly locally uniformly rotund if and only if the norm in $X$ is equi-absolutely continuous. The dual of the product space $(\bigoplus\nolimits_{i=1}^{\infty}X_i)_{\Phi}$ of a sequence of Banach spaces $(X_i)_{i=1}^{\infty}$, which is built by using an Orlicz function $\Phi$ satisfying the $\Delta_2$-condition, is computed isometrically (i.e. the exact norm in the dual is calculated). It is also shown that for any Orlicz function $\Phi$ and any finite system $X_1,\dots,X_n$ of Banach spaces, we have $\mathop WCS((\bigoplus\nolimits_{i=1}^nX_i)_{\Phi})=\min\{\mathop WCS(X_i) i=1,\dots,n\}$ and that if $\Phi$ does not satisfy the $\Delta_2$-condition, then WCS$((\bigoplus\nolimits_{i=1}^{\infty}X_i) _{\Phi})=1$ for any infinite sequence $(X_i)$ of Banach spaces. (English) |
| Keyword:
|
Köthe sequence space |
| Keyword:
|
weakly convergent sequence coefficient |
| Keyword:
|
order continuity of the norm |
| Keyword:
|
absolute continuity of the norm |
| Keyword:
|
compact local uniform rotundity |
| Keyword:
|
Orlicz sequence space |
| Keyword:
|
Luxemburg norm |
| Keyword:
|
Orlicz norm |
| Keyword:
|
dual space |
| Keyword:
|
product space |
| MSC:
|
46A45 |
| MSC:
|
46B20 |
| MSC:
|
46B25 |
| MSC:
|
46B45 |
| MSC:
|
46E20 |
| MSC:
|
46E40 |
| idZBL:
|
Zbl 0941.46005 |
| idMR:
|
MR1780699 |
| DOI:
|
10.21136/MB.1999.126253 |
| . |
| Date available:
|
2009-09-24T21:38:19Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/126253 |
| . |
| Reference:
|
[1] J. M. Ayerbe T. Dominguez Benavides G. Lopez Acedo: Compactness Conditions in Metric Fixed Point Theory.OTAA, vol. 99, Birkhäuser, Basel, 1997. |
| Reference:
|
[2] M. S. Brodskij D. P. Milman: On the center of convex set.Dokl. Akad. Nauk 59 (1948), 837-840. (In Russian.) MR 0024073 |
| Reference:
|
[3] W. I. Bynum: Normal structure coefficient for Banach spaces.Pacific J. Math. 86 (1980), 427-436. MR 0590555, 10.2140/pjm.1980.86.427 |
| Reference:
|
[4] S. T. Chen: Geometry of Orlicz Spaces.Dissertationes Math. 356, 1996. Zbl 1089.46500, MR 1410390 |
| Reference:
|
[5] T. Dominguez Benavides: Weak uniform normal structure in direct sum spaces.Studia Math. 103 (1992), no. 3, 283-290. Zbl 0810.46015, MR 1202012, 10.4064/sm-103-3-283-290 |
| Reference:
|
[6] P. Foralewski H. Hudzik: Some basic properties of generalized Calderón-Lozanovskij spaces.Collect. Math. 48 (1997), no. 4-6. 523-538. MR 1602584 |
| Reference:
|
[7] K. Goebel W. Kirk: Topics in Metrix Fixed Point Theory.Cambridge University Press, Cambridge, 1991. MR 1074005 |
| Reference:
|
[8] A. Kamińska: Flat Orlicz-Musielak sequence spaces.Bull. Polish Acad. Sci. Math. 30 (1982), no. 7-8, 347-352. MR 0707748 |
| Reference:
|
[9] L. V. Kantorovich G. P. Akilov: Functional Analysis.Nauka, Moscow, 1977. (In Russian.) MR 0511615 |
| Reference:
|
[10] T. Landes: Permanence properties of normal structure.Pacific J. Math. 110 (1984), 125-143. Zbl 0534.46015, MR 0722744, 10.2140/pjm.1984.110.125 |
| Reference:
|
[11] T. C. Lin: On normal structure coefficient and the bounded sequence coefficient.Proc. Amer. Math. Soc. 88 (1983), 262-267. MR 0695255, 10.1090/S0002-9939-1983-0695255-2 |
| Reference:
|
[12] J. Lindenstrauss L. Tzafriri: Classical Banach Spaces I.Springer-Verlag, Berlin, 1977. MR 0500056 |
| Reference:
|
[13] W. A. J. Luxemburg: Banach Function Spaces.Thesis, Delft, 1955. Zbl 0068.09204, MR 0072440 |
| Reference:
|
[14] L. Maligranda: Orlicz Spaces and Interpolation.Seminars in Math. 5, Campinas, 1989. Zbl 0874.46022, MR 2264389 |
| Reference:
|
[15] E. Maluta: Uniformly normal structure and related coefficients.Pacific J. Math. 111 (1984), 357-369. Zbl 0495.46012, MR 0734861, 10.2140/pjm.1984.111.357 |
| Reference:
|
[16] J. Musielak: Orlicz Spaces and Modular Spaces.Lecture Notes in Math. 1034, Springer-Verlag, Berlin, 1983. Zbl 0557.46020, MR 0724434 |
| Reference:
|
[17] B. B. Panda O. P. Kapoor: A generalization of local uniform convexity of the norm.J. Math. Anal. Appl. 52 (1975), 300-308. MR 0380365, 10.1016/0022-247X(75)90098-0 |
| Reference:
|
[18] B. Prus: On Bynum's fixed point theorem.Atti Sem. Mat. Fis. Univ. Modena 38 (1990), 535-545. Zbl 0724.46020, MR 1076471 |
| Reference:
|
[19] M. M. Rao Z. D. Ren: Theory of Orlicz Spaces.Marcel Dekker Inc., New York, 1991. MR 1113700 |
| Reference:
|
[20] A. E. Taylor D. C. Lay: Introduction to Functional Analysis.John Wiley & Sons, New York (second edition), 1980. MR 0564653 |
| Reference:
|
[21] G. L. Zhang: Weakly convergent sequence coefficient of product space.Proc. Amer. Math. Soc. 117 (1993), no. 3, 637-643. Zbl 0787.46021, MR 1152993, 10.1090/S0002-9939-1993-1152993-1 |
| . |
