Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
compact set; weak topology; Banach space; dual space; Orlicz sequence spaces
compact set; weak topology; Banach space; dual space; Orlicz sequence spaces
Summary:
We combine the techniques of sequence spaces and general Orlicz functions that are broader than the classical cases of $N$-functions. We give three criteria for the weakly compact sets in general Orlicz sequence spaces. One criterion is related to elements of dual spaces. Under the restriction of $\lim _{u\rightarrow 0} M(u)/u=0$, we propose two other modular types that are convenient to use because they get rid of elements of dual spaces. Subsequently, by one of these two modular criteria, we see that a set $A$ in Riesz spaces $l_p$ $(1 < p < \infty )$ is relatively sequential weakly compact if and only if it is normed bounded, that says $\sup _{u\in A}\sum _{i=1}^{\infty } |u(i)|^p < \nobreak \infty $. The result again confirms the conclusion of the Banach-Alaoglu \hbox {theorem}.
References:
[1] Aleksandrov, P. S., Kolmogorov, A. N.: Introduction to the Theory of Sets and the Theory of Functions. 1. Introduction to the General Theory of Sets and Functions. Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow (1948), Russian. MR 0040369 | Zbl 0037.03404
[2] Andô, T.: Weakly compact sets in Orlicz spaces. Can. J. Math. 14 (1962), 170-176. DOI 10.4153/CJM-1962-012-7 | MR 0157228 | Zbl 0103.32902
[3] Arzelà, C.: Funzioni di linee. Rom. Acc. L. Rend. (4) 5 (1889), 342-348 Italian \99999JFM99999 21.0424.01.
[4] Batt, J., Schlüchtermann, G.: Eberlein compacts in $L_1(X)$. Stud. Math. 83 (1986), 239-250. DOI 10.4064/sm-83-3-239-250 | MR 0850826 | Zbl 0555.46014
[5] Brouwer, L. E. J.: Beweis der Invarianz des $n$-dimensionalen Gebiets. Math. Ann. 71 (1911), 305-313 German \99999JFM99999 42.0418.01. DOI 10.1007/BF01456846 | MR 1511658
[6] Cheng, L., Cheng, Q., Shen, Q., Tu, K., Zhang, W.: A new approach to measures of noncompactness of Banach spaces. Stud. Math. 240 (2018), 21-45. DOI 10.4064/sm8448-2-2017 | MR 3719465 | Zbl 06828572
[7] Cheng, L., Cheng, Q., Zhang, J.: On super fixed point property and super weak compactness of convex subsets in Banach spaces. J. Math. Anal. Appl. 428 (2015), 1209-1224. DOI 10.1016/j.jmaa.2015.03.061 | MR 3334975 | Zbl 1329.46015
[8] Darbo, G.: Punti uniti in trasformazioni a codominio non compatto. Rend. Sem. Mat. Univ. Padova 24 (1955), 84-92 Italian. MR 0070164 | Zbl 0064.35704
[9] Dodds, P. G., Sukochev, F. A., Schlüchtermann, G.: Weak compactness criteria in symmetric spaces of measurable operators. Math. Proc. Camb. Philos. Soc. 131 (2001), 363-384. DOI 10.1017/S0305004101005114 | MR 1857125 | Zbl 1004.46038
[10] Fabian, M., Montesinos, V., Zizler, V.: On weak compactness in $L_1$ spaces. Rocky Mt. J. Math. 39 (2009), 1885-1893. DOI 10.1216/RMJ-2009-39-6-1885 | MR 2575884 | Zbl 1190.46011
[11] Foralewski, P., Hudzik, H., Kolwicz, P.: Non-squareness properties of Orlicz-Lorentz sequence spaces. J. Funct. Anal. 264 (2013), 605-629. DOI 10.1016/j.jfa.2012.10.014 | MR 2997393 | Zbl 1263.46015
[12] Gale, D.: The game of Hex and the Brouwer fixed-point theorem. Am. Math. Mon. 86 (1979), 818-827. DOI 10.2307/2320146 | MR 0551501 | Zbl 0448.90097
[13] James, R. C.: Weakly compact sets. Trans. Am. Math. Soc. 113 (1964), 129-140. DOI 10.1090/S0002-9947-1964-0165344-2 | MR 0165344 | Zbl 0129.07901
[14] James, R. C.: The Eberlein-Šmulian theorem. Functional Analysis. Selected Topics Narosa Publishing House, New Delhi (1998), 47-49. MR 1668790 | Zbl 1010.46011
[15] Kelley, J. L.: General Topology. The University Series in Higher Mathematics. D. van Nostrand, New York (1955). MR 0070144 | Zbl 0066.16604
[16] Kolmogorov, A. N., Tikhomirov, V. M.: $\epsilon$-entropy and $\epsilon$-capacity of sets in function spaces. Usp. Mat. Nauk 14 (1959), 3-86 Russian. MR 0112032 | Zbl 0090.33503
[17] Krasnosel'skij, M. A., Rutitskij, Y. B.: Convex Functions and Orlicz Spaces. P. Noordhoff, Groningen (1961). MR 0126722 | Zbl 0095.09103
[18] Leray, J., Schauder, J.: Topologie et équations fonctionnelles. Ann. Sci. Éc. Norm. Supér., III. Ser. 51 (1934), 45-78 French. DOI 10.24033/asens.836 | MR 1509338 | Zbl 0009.07301
[19] Musielak, J.: Orlicz Spaces and Modular Spaces. Lecture Notes in Mathematics 1034. Springer, Berlin (1983). DOI 10.1007/BFb0072210 | MR 0724434 | Zbl 0557.46020
[20] Schlüchtermann, G.: Weak compactness in $L_\infty(\mu,X)$. J. Funct. Anal. 125 (1994), 379-388. DOI 10.1006/jfan.1994.1129 | MR 1297673 | Zbl 0828.46036
[21] Shi, S., Shi, Z.: On generalized Young's inequality. Function Spaces XII Banach Center Publications 119. Polish Academy of Sciences, Institute of Mathematics. Warsaw (2019), 295-309. DOI 10.4064/bc119-17 | Zbl 1444.46024
[22] Shi, Z., Wang, Y.: Uniformly non-square points and representation of functionals of Orlicz-Bochner sequence spaces. Rocky Mt. J. Math. 48 (2018), 639-660. DOI 10.1216/RMJ-2018-48-2-639 | MR 3810210 | Zbl 1402.46013
[23] Šmulian, V.: On compact sets in the space of measurable functions. Mat. Sb., N. Ser. 15 (1944), 343-346 Russian. MR 0012210 | Zbl 0060.27602
[24] Wu, Y.: Normed compact sets and weakly $H$-compact in Orlicz space. J. Nature 3 (1982), 234 Chinese.
[25] Wu, C., Wang, T.: Orlicz Spaces and Applications. Heilongjiang Sci. & Tech. Press, Harbin (1983), Chinese.
[26] Zhang, X., Shi, Z.: A criterion of compact set in Orlicz sequence space $l_{(M)}$. J. Changchun Post Telcommunication Institute 15 (1997), 64-67 Chinese.
Search
Partner of
