On Perturbed Isometries Between the Positive Cones of Certain Continuous Function Spaces

Let X , Y be two compact Hausdorff perfectly normal spaces (in particular, compact metrizable spaces), C ( X ) be the real Banach space of all continuous functions on X , and C + ( X ) be the positive cone of C ( X ). In this paper, we show that if there exists a δ -surjective ε -isometry F : C + (...

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Veröffentlicht in:	Resultate der Mathematik Jg. 78; H. 2; S. 63
Hauptverfasser:	Sun, Longfa, Sun, Yinghua, Wang, Shenghua
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Cham Springer International Publishing 01.04.2023 Springer Nature B.V
Schlagworte:	Approximation Banach spaces Continuity (mathematics) Function space Mathematical functions Mathematicians Mathematics Mathematics and Statistics Primary 46B04 Banach–Stone theorem Isometries Continuous function space 46B20 Secondary 46E15 Hyers–Ulam stability
ISSN:	1422-6383, 1420-9012
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Abstract	Let X , Y be two compact Hausdorff perfectly normal spaces (in particular, compact metrizable spaces), C ( X ) be the real Banach space of all continuous functions on X , and C + ( X ) be the positive cone of C ( X ). In this paper, we show that if there exists a δ -surjective ε -isometry F : C + ( X ) → C + ( Y ) , then X and Y are homeomorphic. Moreover, we show that there exists a unique additive surjective isometry V : C + ( X ) → C + ( Y ) (the restriction of a linear surjective isometry U : C ( X ) → C ( Y ) induced by the homeomorphism) such that ‖ F ( f ) - V ( f ) ‖ ≤ 2 ε , for all f ∈ C + ( X ) . This can be regarded as a localized generalization of the Banach–Stone theorem for compact Hausdorff perfectly normal spaces.
AbstractList	Let X , Y be two compact Hausdorff perfectly normal spaces (in particular, compact metrizable spaces), C ( X ) be the real Banach space of all continuous functions on X , and C + ( X ) be the positive cone of C ( X ). In this paper, we show that if there exists a δ -surjective ε -isometry F : C + ( X ) → C + ( Y ) , then X and Y are homeomorphic. Moreover, we show that there exists a unique additive surjective isometry V : C + ( X ) → C + ( Y ) (the restriction of a linear surjective isometry U : C ( X ) → C ( Y ) induced by the homeomorphism) such that ‖ F ( f ) - V ( f ) ‖ ≤ 2 ε , for all f ∈ C + ( X ) . This can be regarded as a localized generalization of the Banach–Stone theorem for compact Hausdorff perfectly normal spaces. Let X, Y be two compact Hausdorff perfectly normal spaces (in particular, compact metrizable spaces), C(X) be the real Banach space of all continuous functions on X, and C+(X) be the positive cone of C(X). In this paper, we show that if there exists a δ-surjective ε-isometry F:C+(X)→C+(Y), then X and Y are homeomorphic. Moreover, we show that there exists a unique additive surjective isometry V:C+(X)→C+(Y) (the restriction of a linear surjective isometry U:C(X)→C(Y) induced by the homeomorphism) such that ‖F(f)-V(f)‖≤2ε,forallf∈C+(X).This can be regarded as a localized generalization of the Banach–Stone theorem for compact Hausdorff perfectly normal spaces.
ArticleNumber	63
Author	Sun, Yinghua Sun, Longfa Wang, Shenghua
Author_xml	– sequence: 1 givenname: Longfa surname: Sun fullname: Sun, Longfa email: sun.longfa@163.com organization: Hebei Key Laboratory of Physics and Energy Technology, School of Mathematics and Physics, North China Electric Power University – sequence: 2 givenname: Yinghua surname: Sun fullname: Sun, Yinghua organization: Hebei Key Laboratory of Physics and Energy Technology, School of Mathematics and Physics, North China Electric Power University – sequence: 3 givenname: Shenghua surname: Wang fullname: Wang, Shenghua organization: Hebei Key Laboratory of Physics and Energy Technology, School of Mathematics and Physics, North China Electric Power University
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Cites_doi	10.1007/BF03322748 10.1090/S0002-9947-1937-1501905-7 10.1007/BF01461008 10.1007/BF02785416 10.1112/S0024609398005591 10.1016/j.jmaa.2020.124014 10.1007/s13226-021-00047-2 10.1007/BF02773375 10.1016/j.jmaa.2015.10.035 10.1016/j.jfa.2016.07.008 10.1007/s11425-016-5155-0 10.1007/s11253-020-01807-9 10.1090/S0002-9904-1945-08337-2 10.1016/S0022-1236(02)00049-6 10.1016/j.jfa.2012.11.008 10.1016/j.jfa.2015.04.015 10.1016/j.na.2005.05.036 10.1090/coll/048
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Snippet	Let X , Y be two compact Hausdorff perfectly normal spaces (in particular, compact metrizable spaces), C ( X ) be the real Banach space of all continuous... Let X, Y be two compact Hausdorff perfectly normal spaces (in particular, compact metrizable spaces), C(X) be the real Banach space of all continuous functions...
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SubjectTerms	Approximation Banach spaces Continuity (mathematics) Function space Mathematical functions Mathematicians Mathematics Mathematics and Statistics
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Title	On Perturbed Isometries Between the Positive Cones of Certain Continuous Function Spaces
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