The Banach-Mazur distance between C(Δ) and C_0(Δ) equals 2
DOI:
https://doi.org/10.12775/TMNA.2023.025Keywords
Banach-Mazur distance, Cantor set, space of continuous functionsAbstract
Let $C(\Delta)$ denote the Banach space of all continuous real-valued functions on the Cantor set $\Delta$ and $C_0(\Delta)=\lbrace f\in C(\Delta): f(1)=0\rbrace$. From the 1966 theorem of Cambern, it is well-known that the Banach-Mazur distance $d(C(\Delta), C_0(\Delta))\geq 2$. We prove that, in fact, $d(C(\Delta), C_0(\Delta))= 2$. As a consequence, we answer a question left open in the 2012 paper of Candido and Galego.References
S. Banach, Théorie des opérations linéaires, Monografie Matematyczne, Warszawa, 1932.
C. Bessaga and A. Pelczyński, Spaces of continuous functions IV, Studia Math. 19 (1960), 53–61.
M. Cambern, A generalized Banach–Stone theorem, Proc. Amer. Math. Soc. 17 (1966), 396–400.
M. Cambern, Isomorphisms of C0 (Y ) onto C(X), Pacific J. Math. 35 (1970), 307–312.
M. Cambern, Isomorphisms of C0 (Y ) with Y discrete, Math. Ann. 188 (1970), 23–25.
M. Cambern, On mappings of sequence spaces, Studia Math. 30 (1968), 73–77.
M. Cambern, On isomorphisms with small bound, Proc. Amer. Math. Soc. 18 (1967), 1062–1066.
L. Candido and E.M. Galego, How far is C0 (Γ, X) with Γ discrete from C0 (K, X) spaces?, Fund. Math. 218 (2012), 151–163.
L. Candido and E.M. Galego, How far is C(ω) from the other C(K) spaces?, Studia Math. 217 (2013), no. 2, 123–138.
A. Gergont and L. Piasecki, On isomorphic embeddings of c into L1 -preduals and some applications, J. Math. Anal. Appl. 492 (2020), no. 1, 124431, 11 pp.
A. Gergont and L. Piasecki, Some topological and metric properties of the space of `1 -predual hyperplanes in c, Colloq. Math. 168 (2022), no. 2, 229–247.
Y. Gordon, On the distance coefficient between isomorphic function spaces, Israel J. Math. 8 (1970), 391–397.
S. Mazurkiewicz and W. Sierpiński, Contribution à la topologie des ensembles dénombrables, Fund. Math. 1 (1920), 17–27.
A.A. Miljutin, Isomorphism of the spaces of continuous functions over compact sets of the cardinality of the continuum, Teor. Funkciı̆ Funkcional. Anal. i Priložen. Vyp. 2 (1966), 150–156. (Russian)
S. Willard, General Topology, Addison–Wesley Publishing Company, 1970.
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Copyright (c) 2024 Łukasz Piasecki, Jeimer Villada
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