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Modular metric spaces. I: Basic concepts. (English) Zbl 1200.54014
The author introduces the notion of a (metric) modular as follows: A (metric) modular on a set $X$ is a function $w:(0,\infty)\times X\times X\rightarrow [0,\infty]$ satisfying, for all $x,y,z\in X,$ the following three properties: $x=y$ if and only if $w(\lambda,x,y)=0$ for all $\lambda >0;$ $w(\lambda,x,y)=w(\lambda,y,x)$ for all $\lambda>0;$ and $w(\lambda+\mu,x,y)\leq w(\lambda,x,z)+w(\mu,y,z)$ for all $\lambda,\mu >0.$ He shows that given $x_0\in X$ the set $X_w=\{x\in X:\lim_{\lambda\to\infty}w(\lambda,x,x_0)=0\}$ is a metric space with metric $d^o_w(x,y)=\inf \{\lambda >0: w(\lambda,x,y)\leq \lambda\},$ which he calls a modular space. The article develops the theory of metric spaces generated by (convex) modulars. In this way the author is able to extend results from Nakano’s theory of modular linear spaces to his setting. Among other things, the method can be used to define new metric spaces of (multivalued) functions of bounded generalized variation of a real variable with values in metric semigroups and abstract convex cones.

MSC:
54E35Metric spaces, metrizability
46A80Modular topological linear spaces
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References:
[1] Musielak, J.; Orlicz, W.: On generalized variations, I, Studia math. 18, 11-41 (1959) · Zbl 0088.26901
[2] Orlicz, W.: Collected papers, vols. I, II, (1988) · Zbl 0675.01024
[3] Orlicz, W.: A note on modular spaces, I, Bull. acad. Polon. sci. Sér. sci. Math. astron. Phys. 9, 157-162 (1961) · Zbl 0109.33404
[4] Musielak, J.; Orlicz, W.: Some remarks on modular spaces, Bull. acad. Polon. sci. Sér. sci. Math. astron. Phys. 7, 661-668 (1959) · Zbl 0099.09202
[5] Musielak, J.; Orlicz, W.: On modular spaces, Studia math. 18, 49-65 (1959) · Zbl 0086.08901
[6] Nakano, H.: Modulared semi-ordered linear spaces, Tokyo math. Book ser. 1 (1950) · Zbl 0041.23401
[7] Nakano, H.: Topology and linear topological spaces, Tokyo math. Book ser. 3 (1951)
[8] Koshi, S.; Shimogaki, T.: On F-norms of quasi-modular spaces, J. fac. Sci. hokkaido univ. Ser. I 15, No. 3--4, 202-218 (1961) · Zbl 0099.31404
[9] Yamamuro, S.: On conjugate spaces of nakano spaces, Trans. amer. Math. soc. 90, 291-311 (1959) · Zbl 0086.09002 · doi:10.2307/1993206
[10] W.A.J. Luxemburg, Banach function spaces, Thesis, Delft, Inst. of Techn., Assen, The Netherlands, 1955 · Zbl 0068.09204
[11] Mazur, S.; Orlicz, W.: On some classes of linear spaces, Studia math. 17, 97-119 (1958) · Zbl 0085.32203
[12] Turpin, Ph.: Fubini inequalities and bounded multiplier property in generalized modular spaces, Comment. math., tomus specialis in honorem ladislai Orlicz 1, 331-353 (1978) · Zbl 0391.46028
[13] Chistyakov, V. V.: Lipschitzian superposition operators between spaces of functions of bounded generalized variation with weight, J. appl. Anal. 6, No. 2, 173-186 (2000) · Zbl 0997.47051 · doi:10.1515/JAA.2000.173
[14] Chistyakov, V. V.: Mappings of generalized variation and composition operators, J. math. Sci. (NY) 110, No. 2, 2455-2466 (2002) · Zbl 1033.26013 · doi:10.1023/A:1015018310969
[15] Gniłka, S.: Modular spaces of functions of bounded M-variation, Funct. approx. Comment. math. 6, 3-24 (1978) · Zbl 0396.46031
[16] Herda, H. -H.: Modular spaces of generalized variation, Studia math. 30, 21-42 (1968) · Zbl 0159.18103
[17] H. Hudzik, L. Maligranda, Amemiya norm equals Orlicz norm in general, Dept. of Math., Luleå Univ. of Technology, Research Report 1999-05, Luleå, Sweden, 1999, pp. 1--15 · Zbl 1010.46031
[18] Leśniewicz, R.; Orlicz, W.: On generalized variations, II, Studia math. 45, 71-109 (1973) · Zbl 0247.26006
[19] Maligranda, L.: Orlicz spaces and interpolation, Seminars in math. 5 (1989) · Zbl 0874.46022
[20] Maligranda, L.; Orlicz, W.: On some properties of functions of generalized variation, Monatsh. math. 104, 53-65 (1987) · Zbl 0623.26009 · doi:10.1007/BF01540525
[21] Musielak, J.: Orlicz spaces and modular spaces, Lecture notes in math. 1034 (1983) · Zbl 0557.46020
[22] Rolewicz, S.: Metric linear spaces, (1985) · Zbl 0573.46001
[23] Schembari, N. P.; Schramm, M.: {$\Phi$}V[h] and Riemann--Stieltjes integration, Colloq. math. 60--61, 421-441 (1990) · Zbl 0741.26007
[24] Adams, R. A.: Sobolev spaces, Pure appl. Math. 65 (1975) · Zbl 0314.46030
[25] Rao, M. M.; Ren, Z. D.: Applications of Orlicz spaces, Pure appl. Math. 250 (2004)
[26] Chistyakov, V. V.: Generalized variation of mappings and applications, Real anal. Exchange 25, 61-64 (1999--2000) · Zbl 1014.26524
[27] V.V. Chistyakov, On mappings of finite generalized variation and nonlinear operators, in: Real Analysis Exchange 24th Summer Symposium Conference Reports, Denton, Texas, USA, 2000, pp. 39--43 · Zbl 1105.26302
[28] Chistyakov, V. V.: Generalized variation of mappings with applications to composition operators and multifunctions, Positivity 5, No. 4, 323-358 (2001) · Zbl 1027.47046 · doi:10.1023/A:1011879221347
[29] Chistyakov, V. V.: On multi-valued mappings of finite generalized variation, Math. notes 71, No. 3--4, 556-575 (2002)
[30] Chistyakov, V. V.: Selections of bounded variation, J. appl. Anal. 10, No. 1, 1-82 (2004) · Zbl 1077.26015
[31] Chistyakov, V. V.: Lipschitzian nemytskii operators in the cones of mappings of bounded Wiener ${\phi}$-variation, Folia math. 11, No. 1, 15-39 (2004) · Zbl 1101.47045
[32] Rådström, H.: An embedding theorem for spaces of convex sets, Proc. amer. Math. soc. 3, No. 1, 165-169 (1952) · Zbl 0046.33304 · doi:10.2307/2032477
[33] Chistyakov, V. V.: Modular metric spaces generated by F-modulars, Folia math. 14, 3-25 (2008) · Zbl 1266.54072
[34] Chistyakov, V. V.: Metric modulars and their application, Dokl. math. 73, No. 1, 32-35 (2006) · Zbl 1155.46304 · doi:10.1134/S106456240601008X
[35] Matuszewska, W.; Orlicz, W.: On property B1 for functions of bounded ${\phi}$-variation, Bull. Polish acad. Sci. math. 35, No. 1--2, 57-69 (1987) · Zbl 0625.46033
[36] Chistyakov, V. V.: Abstract superposition operators on mappings of bounded variation of two real variables. I, Siberian math. J. 46, No. 3, 555-571 (2005) · Zbl 1097.26010 · emis:journals/SMZ/2005/03/698.htm
[37] Smajdor, W.: Note on Jensen and pexider functional equations, Demonstratio math. 32, No. 2, 363-376 (1999) · Zbl 0938.39026
[38] Hörmander, L.: Sur la fonction d’appui des ensembles convexes dans un espace localement convexe, Ark. mat. 3, No. 12, 181-186 (1954) · Zbl 0064.10504 · doi:10.1007/BF02589354
[39] Pinsker, A. G.: The space of convex sets of a locally convex space, Collection of papers of leningrad. Engineer.-econom. Inst. named after P. Togliatti 63, 13-17 (1966)
[40] De Blasi, F. S.: On the differentiability of multifunctions, Pacific J. Math. 66, No. 1, 67-81 (1976) · Zbl 0348.58004
[41] Castaing, C.; Valadier, M.: Convex analysis and measurable multifunctions, Lecture notes in math. 580 (1977) · Zbl 0346.46038