Abstract
In this paper, we obtain that every isometry from the unit sphere S(l p(Γ)) of l p(Γ) (1 < p < ∞, p ≠ 2) onto the unit sphere S(E) of a Banach space E can be extended to be a (real) linear isometry of l p(Γ) onto E, so, we give an affirmative answer to the corresponding Tingley’s problem.
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References
Ding G G. The 1-Lipschitz mappings between the unit spheres of two Hilbert spaces can be extended to a real linear isometry of the whole space. Sci China Ser A, 2002, 45: 479–483
Ding G G. On the extension of isometries between unit spheres of E and C(Ω). Acta Math Sinica Engl Ser, 2003, 19: 793–800
Ding G G. The isometric extension problem in the unit spheres of l p(Γ) (p > 1) type spaces. Sci China Ser A, 2003, 46: 333–338
Ding G G. The representation theorem of onto isometric mappings between two spheres of l ∞ type spaces and the application on isometric extension problem. Sci China Ser A, 2004, 47: 722–729
Ding G G. The representation theorem of onto isometric mappings between two unit spheres of l 1(Γ) type spaces and the application to the isometric extension problem. Acta Math Sinica, 2004, 20: 1089–1094
Ding G G. The isometric extension of the into mapping from a \( \mathcal{L} \) ∞(Γ)-type space to some Banach space. Illinois J Math, 2007, 51: 445–453
Ding G G. Extension of isometries on the unit sphere of AL-space. Sci China Ser A, 2008, 38: 541–555
Ding G G. On isometric extension problem between two unit spheres. Sci China Ser A, 2009, 52: 2069–2083
Fang X N, Wang J H. On linear extension of isometries between the unit spheres. Acta Math Sinica Engl Ser, 2005, 48: 1109–1112
Fang X N, Wang J H. Extension of isometries between the unit spheres of normed space E and C(Ω). Acta Math Sinica Engl Ser, 2006, 22: 1819–1824
Fang X N, Wang J H. Extension of isometries between unit spheres of normed space E and l 1(Γ). Acta Math Sinica Engl Ser, 2008, 51: 24–28
Liu R. On extension of isometries between unit spheres of \( \mathcal{L} \) ∞(Γ)-type space and a Banach space E. J Math Anal Appl, 2007, 333: 959–970
Mankiewicz P. On extension of isometries in normed linear spaces. Bull Acad Polon Sci Ser Sci Math Astronom Phys, 1972, 20: 367–371
Mazur S, Ulam S. Sur less transformations isométriques d’espaces vectoriels normés. C R Acad Sci Paris, 1932, 194: 946–948
Royden H L. Real Analysis. New York: Macmillan, 1968
Tingley D. Isometries of the unit sphere. Geom Dedicata, 1987, 22: 371–378
Wang J. On extension of isometries between unit spheres of AL p-spaces (0 < p < ∞). Proc Amer Math Soc, 2004, 132: 2899–2909
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Fang, X., Wang, J. Extension of isometries on the unit sphere of l p(Γ) space. Sci. China Math. 53, 1085–1096 (2010). https://doi.org/10.1007/s11425-010-0028-4
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DOI: https://doi.org/10.1007/s11425-010-0028-4