On the smoothness of normed spaces. (English) Zbl 1537.46012
This (mostly) survey on smoothness and convexity in normed spaces \(E\) starts out very elementarily with the basic connection between smoothness and strict convexity. In Definition 2.8, a simplified formulation of uniform convexity is presented.
Section 3 is devoted to the usual modulus of convexity, \(\delta_E\), while in Section 4 a modulus of smoothness, \(\bar{\rho}_E\), slightly different from Day’s classical modulus of smoothness, is presented with its properties.
The authors call the (non-negative) difference \(d_E(\varepsilon)=\bar{\rho}_E(\varepsilon)-\delta_E(\varepsilon)\), \(\varepsilon\in[0,2]\), the modulus of deformation of \(E\) while the deformation of \(E\) is the supremum of \(d_E\) over the intervall \([0,2]\). It turns out that Hilbert spaces have zero deformation while \(d_{C[a,b]}=1\), the biggest possible deformation among normed spaces.
Section 3 is devoted to the usual modulus of convexity, \(\delta_E\), while in Section 4 a modulus of smoothness, \(\bar{\rho}_E\), slightly different from Day’s classical modulus of smoothness, is presented with its properties.
The authors call the (non-negative) difference \(d_E(\varepsilon)=\bar{\rho}_E(\varepsilon)-\delta_E(\varepsilon)\), \(\varepsilon\in[0,2]\), the modulus of deformation of \(E\) while the deformation of \(E\) is the supremum of \(d_E\) over the intervall \([0,2]\). It turns out that Hilbert spaces have zero deformation while \(d_{C[a,b]}=1\), the biggest possible deformation among normed spaces.
Reviewer: Olav Nygaard (Kristiansand)
MSC:
| 46B20 | Geometry and structure of normed linear spaces |
| 46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |
Keywords:
normed space; strict convexity; uniform convexity; smoothness; uniform smoothness; modulus of convexity; modulus of smoothness; modulus of deformationReferences:
| [1] | Ahmad, A.; Fu, Y.; Li, Y., Some properties concerning the \(J_L(X)\) and \(Y_J(X)\) which related to some special inscribed triangles of unit ball, Symmetry, 13, 1285 (2021) · doi:10.3390/sym13071285 |
| [2] | Alonso, J.; Ullan, A., Moduli in normed linear spaces and characterization of inner product spaces, Arch. Math., 50, 487-495 (1992) · Zbl 0760.46014 · doi:10.1007/BF01236044 |
| [3] | Banaś, J., On modulus of smoothness of Banach spaces, Bull. Pol. Acad. Sci. Math., 34, 287-293 (1986) · Zbl 0606.46010 |
| [4] | Banaś, J.; Frączek, K., Deformation of Banach spaces, Comment. Math. Univ. Carolinae, 34, 47-53 (1993) · Zbl 0817.46016 |
| [5] | Banaś, J.; Rzepka, B., Functions related to convexity and smoothness of normed spaces, Rend. Circ. Mat. Palermo, 46, 395-424 (1997) · Zbl 0913.46014 · doi:10.1007/BF02844281 |
| [6] | Banaś, J.; Hajnosz, A.; Wędrychowiczz, S., On convexity and smoothness of Banach spaces, Comment. Math. Univ. Carolinae, 31, 445-452 (1990) · Zbl 0723.46010 |
| [7] | Baronti, M.; Papini, PL, Convexity, smoothness and moduli, Nonlinear Anal., 70, 2457-2465 (2009) · Zbl 1175.46012 · doi:10.1016/j.na.2008.03.030 |
| [8] | Beauzamy, B.: Introduction to Banach Spaces and Their Geometry, Notas de Matemática 86, Mathematics Studies, vol. 68. North-Holland, Amsterdam (1982) · Zbl 0491.46014 |
| [9] | Bernstein, F.; Doetsch, G., Zur Theorie der konvexen Funktionen, Math. Ann., 76, 514-526 (1915) · JFM 45.0627.02 · doi:10.1007/BF01458222 |
| [10] | Clarkson, J., Uniformly convex spaces, Trans. Am. Math. Soc., 40, 396-414 (1936) · JFM 62.0460.04 · doi:10.1090/S0002-9947-1936-1501880-4 |
| [11] | Cui, H.; Zhang, Y., A note on the Banaś modulus of smoothness in the Bynum space, Appl. Math. Lett., 23, 299-301 (2010) · Zbl 1194.46022 · doi:10.1016/j.aml.2009.10.002 |
| [12] | Daneš, J., On Densifying and Related Mappings and Their Applications in Nonlinear Functional Analysis, Theory of Nonlinear Operators, 15-56 (1974), Berlin: Akademie Verlag, Berlin · Zbl 0295.47058 |
| [13] | Day, MM, Uniform convexity in factor and conjugate spaces, Ann. Math., 45, 375-385 (1944) · Zbl 0063.01058 · doi:10.2307/1969275 |
| [14] | Day, MM, Strict convexity and smoothness of normed spaces, Trans. Am. Math. Soc., 78, 516-528 (1955) · Zbl 0068.09101 · doi:10.1090/S0002-9947-1955-0067351-1 |
| [15] | Day, MM, Normed Linear Spaces (1973), Berlin: Springer, Berlin · Zbl 0268.46013 · doi:10.1007/978-3-662-09000-8 |
| [16] | Du, D.; Ahmad, A.; Din, A.; Li, Y., Some moduli of angles in Banach spaces, Mathematics, 10, 2965 (2022) · doi:10.3390/math10162965 |
| [17] | Dunford, N.; Schwartz, JT, Linear Operators I (1963), Leyden: Int. Publ., Leyden |
| [18] | Figiel, T., On moduli of convexity and smoothness, Stud. Math., 56, 121-155 (1976) · Zbl 0344.46052 · doi:10.4064/sm-56-2-121-155 |
| [19] | Gao, J., Normal structure and some parameters in Banach spaces, Nonlinear Funct. Anal. Appl., 10, 299-310 (2005) · Zbl 1092.46011 |
| [20] | Goebel, K., Convexity of balls and fixed point theorems for mappings with nonexpansive square, Compos. Math., 22, 269-274 (1970) · Zbl 0202.12802 |
| [21] | Goebel, K.; Kirk, WA, Topics in Metric Fixed Point Theory (1990), Cambridge: Cambridge Univeristy Press, Cambridge · Zbl 0708.47031 · doi:10.1017/CBO9780511526152 |
| [22] | Goebel, K.; Prus, S., Elements of Geometry of Balls in Banach Spaces (2018), Oxford: Oxford University Press, Oxford · Zbl 1430.46001 · doi:10.1093/oso/9780198827351.001.0001 |
| [23] | Gurarii, VI, On the differentiability properties of the moduli of rotundity of Banach spaces, Mat. Issled., 2, 141-148 (1967) · Zbl 0232.46024 |
| [24] | Ivanov, G.; Martini, H., New moduli for Banach spaces, Ann. Funct. Anal., 8, 350-365 (2017) · Zbl 1379.46018 · doi:10.1215/20088752-2017-0001 |
| [25] | Jang, C.; Li, H., Generalized von Neumann-Jordan constant for the Banaś-Frączek space, Colloq. Math., 154, 149-156 (2018) · Zbl 1410.46008 · doi:10.4064/cm7422-1-2018 |
| [26] | Köthe, G., Topological Vector Spaces I (1969), Berlin: Springer, Berlin · Zbl 0179.17001 |
| [27] | Lindenstrauss, J., On the modulus of smoothness and divergent series in Banach spaces, Mich. Math. J., 10, 241-252 (1963) · Zbl 0115.10001 · doi:10.1307/mmj/1028998906 |
| [28] | Lindenstrauss, J.; Tzafriri, L., Classical Banach Spaces (1973), Berlin: Springer, Berlin · Zbl 0262.46031 |
| [29] | Liokoumovich, VI, The existence of \(B\)-spaces with nonconvex modulus of convexity (in Russian), Izv. Vyss. Uchebn. Zaved. Math., 12, 43-49 (1973) · Zbl 0274.46014 |
| [30] | Llorens Fuster, E.: Some moduli and constants related to metric fixed point theory. In: Kirk, W.A., et al. (ed.), Handbook of Metric Fixed Point Theory, pp. 133-175. Kluwer Academic Publishers, Dordrecht (2001) · Zbl 1021.47030 |
| [31] | Milman, VD, The geometric theory of Banach spaces, part II, Usp. Mat. Nauk, 26, 73-149 (1971) · Zbl 0229.46017 |
| [32] | Mitani, KI; Saito, KS; Takahashi, Y., On the von Neumann-Jordan constant of generalized Banaś-Frączek spaces, Linear Nonlinear Anal., 2, 311-316 (2016) · Zbl 1424.46018 |
| [33] | Nordlander, G., The modulus of convexity in normed linear spaces, Ark. Math., 4, 15-17 (1960) · Zbl 0092.11402 · doi:10.1007/BF02591317 |
| [34] | Prus, S.; Stachura, A., Functional Analysis in Exercises (in Polish) (2007), Warszawa: Wydawnictwo Naukowe PWN, Warszawa |
| [35] | Rainwater, J., Local uniform convexity of Days norm on \(c_0(\Gamma )\), Proc. Am. Math. Soc., 22, 335-339 (1969) · Zbl 0185.37602 |
| [36] | Saejung, S.; Gao, J., On Banaś-Hajnosz-Wędrychowicz type modulus of convexity and fixed point property, Nonlinear Funct. Anal. Appl., 21, 717-725 (2016) · Zbl 1431.46008 |
| [37] | Sain, D.; Ghosh, S.; Paul, K., On isosceles orthogonality and some geometric constants in a normed space, Aequ. Math., 97, 147-160 (2023) · Zbl 1520.46005 · doi:10.1007/s00010-022-00909-y |
| [38] | Sierpiński, W., Sur les fonctions convexes mesurables, Fund. Math., 1, 125-128 (1920) · JFM 47.0235.03 · doi:10.4064/fm-1-1-125-128 |
| [39] | Ullan, A.: Moduli of Convexity and Smoothness in Normed Spaces (in Spanish). Ph. D. Thesis, Extremadura University, Badajoz (1990) |
| [40] | Yang, C., Jordan-von Neumann constant for Banaś-Frączek space, Banach J. Math. Anal., 8, 185-192 (2014) · Zbl 1304.46019 · doi:10.15352/bjma/1396640062 |
| [41] | Yang, C.; Yang, X., On the James type constant and von Neumann-Jordan constant for a class of Banaś-Frączek type spaces, J. Math. Inequal., 10, 551-558 (2016) · Zbl 1348.46017 · doi:10.7153/jmi-10-43 |
| [42] | Zuo, ZF, Banaś-Hajnosz-Wędrychowicz type modulus of convexity and normal structure in Banach spaces, J. Fixed Point Theory Appl., 20, 58, 139 (2018) · Zbl 1407.46014 |
| [43] | Zuo, Z.; Cui, Y., Some modulus and normal structure in Banach space, J. Inequal. Appl., 2009 (2009) · Zbl 1178.46013 · doi:10.1155/2009/676373 |
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