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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| Vydáno v: | Resultate der Mathematik Ročník 78; číslo 2; s. 63 |
|---|---|
| Hlavní autoři: | , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Cham
Springer International Publishing
01.04.2023
Springer Nature B.V |
| Témata: | |
| ISSN: | 1422-6383, 1420-9012 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | 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. |
|---|---|
| Bibliografie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 1422-6383 1420-9012 |
| DOI: | 10.1007/s00025-023-01844-3 |