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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| Published in: | Resultate der Mathematik Vol. 78; no. 2; p. 63 |
|---|---|
| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
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Springer International Publishing
01.04.2023
Springer Nature B.V |
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| 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 |
| ContentType | Journal Article |
| Copyright | The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023. |
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| Keywords | Primary 46B04 Banach–Stone theorem Isometries Continuous function space 46B20 Secondary 46E15 Hyers–Ulam stability |
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| References | Mazur, Ulam (CR12) 1932; 194 Vestfrid (CR21) 2005; 63 Zhou, Zhang, Liu (CR25) 2018; 44 Dilworth (CR7) 1999; 31 Dugundji (CR8) 1966 Cheng, Dong, Zhang (CR5) 2013; 264 Alestalo, Trotsenko, Väisälä (CR1) 2001; 125 Vestfrid (CR23) 2020; 72 Zhou, Zhang, Liu (CR24) 2016; 435 Cheng, Cheng, Tu, Zhang (CR4) 2015; 269 Stone (CR16) 1937; 41 Väisälä (CR20) 2003; 43 Banach (CR2) 1963 Sun (CR17) 2020 CR3 Cheng, Shen, Zhang, Zhou (CR6) 2017; 60 Phelps (CR14) 1993 Galego, da Silva (CR9) 2016; 271 Garrido, Jaramillo (CR10) 2002; 17 Sun (CR18) 2021 Hyers, Ulam (CR11) 1945; 51 Šemrl, Väisälä (CR15) 2003; 198 Väisälä (CR19) 2002; 128 Vestfrid (CR22) 2007; 12 Omladič, Šemrl (CR13) 1995; 303 S Mazur (1844_CR12) 1932; 194 IA Vestfrid (1844_CR23) 2020; 72 P Alestalo (1844_CR1) 2001; 125 IA Vestfrid (1844_CR22) 2007; 12 L Cheng (1844_CR4) 2015; 269 EM Galego (1844_CR9) 2016; 271 IA Vestfrid (1844_CR21) 2005; 63 S Banach (1844_CR2) 1963 P Šemrl (1844_CR15) 2003; 198 L Sun (1844_CR17) 2020 M Omladič (1844_CR13) 1995; 303 L Cheng (1844_CR6) 2017; 60 J Väisälä (1844_CR19) 2002; 128 J Väisälä (1844_CR20) 2003; 43 L Cheng (1844_CR5) 2013; 264 MH Stone (1844_CR16) 1937; 41 Y Zhou (1844_CR24) 2016; 435 Y Zhou (1844_CR25) 2018; 44 SJ Dilworth (1844_CR7) 1999; 31 DH Hyers (1844_CR11) 1945; 51 J Dugundji (1844_CR8) 1966 R Phelps (1844_CR14) 1993 1844_CR3 L Sun (1844_CR18) 2021 MI Garrido (1844_CR10) 2002; 17 |
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Soc. doi: 10.1090/S0002-9947-1937-1501905-7 – volume-title: Topology year: 1966 ident: 1844_CR8 – volume: 43 start-page: 359 year: 2003 ident: 1844_CR20 publication-title: Result Math. doi: 10.1007/BF03322748 – volume: 60 start-page: 277 issue: 2 year: 2017 ident: 1844_CR6 publication-title: Sci. China Math. doi: 10.1007/s11425-016-5155-0 – volume: 435 start-page: 754 year: 2016 ident: 1844_CR24 publication-title: J. Math. Anal. App. doi: 10.1016/j.jmaa.2015.10.035 – volume-title: Théorie des Opérations Lineaires, Reprinted year: 1963 ident: 1844_CR2 – volume: 198 start-page: 268 year: 2003 ident: 1844_CR15 publication-title: J. Funct. Anal. doi: 10.1016/S0022-1236(02)00049-6 – volume: 271 start-page: 2166 year: 2016 ident: 1844_CR9 publication-title: J. Funct. Anal. doi: 10.1016/j.jfa.2016.07.008 – volume-title: Convex Functions, Monotone Operators and Differentiability. Lecture Notes in Mathematics year: 1993 ident: 1844_CR14 – volume: 269 start-page: 199 issue: 1 year: 2015 ident: 1844_CR4 publication-title: J. Funct. Anal. doi: 10.1016/j.jfa.2015.04.015 – volume: 125 start-page: 61 year: 2001 ident: 1844_CR1 publication-title: Isr. J. Math. doi: 10.1007/BF02773375 – volume: 17 start-page: 351 year: 2002 ident: 1844_CR10 publication-title: Extracta Math. – volume: 128 start-page: 1 year: 2002 ident: 1844_CR19 publication-title: Isr. J. Math. doi: 10.1007/BF02785416 – volume: 303 start-page: 617 year: 1995 ident: 1844_CR13 publication-title: Math. Ann. doi: 10.1007/BF01461008 – volume: 63 start-page: 1191 year: 2005 ident: 1844_CR21 publication-title: Nonlinear Anal. doi: 10.1016/j.na.2005.05.036 – volume: 264 start-page: 713 issue: 3 year: 2013 ident: 1844_CR5 publication-title: J. Funct. Anal. doi: 10.1016/j.jfa.2012.11.008 |
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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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| StartPage | 63 |
| 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 |
| URI | https://link.springer.com/article/10.1007/s00025-023-01844-3 https://www.proquest.com/docview/3256379530 |
| Volume | 78 |
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