Abstract
LetC denote the Banach space of scalar-valued continuous functions defined on the closed unit interval. It is proved that ifX is a Banach space andT:C→X is a bounded linear operator withT * X * non-separable, then there is a subspaceY ofC, isometric toC, such thatT|Y is an isomorphism. An immediate consequence of this and a result of A. Pelczynski, is that every complemented subspace ofC with non-separable dual is isomorphic (linearly homeomorphic) toC.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. Dugundji,An extension of Tietze’s theorem, Pacific J. Math.1 (1951), 353–367.
J. Hagler,Some more Banach spaces which contain l 1, to appear, Studia Math.
J. Hagler and C. Stegall,Banach spaces whose duals contain complemented subspaces isomorphic to C [0, 1]*, preprint.
W. B. Johnson and H. P. Rosenthal,On w *-basic sequences and their applications to the study of Banach spaces, Studia Math.43 (1972), 77–92.
J. Lindenstrauss and A. Pelcynski,Contributions to the theory of the classical Banach spaces, J. Functional Analysis8 (1971), 225–249.
A. A. Milutin,Isomorphism of spaces of continuous functions on compacts of the power continuum, Teor. Funkcional. Anal. i Prilozen.2 (1966), 150–156 (Russian).
A. Pelczynski,Linear extensions, linear averagings and their application to linear topological classification of spaces of continuous functions, Rozprawy Matematyczne58 (1968).
A. Pelczynski,On Banach space containing L 1(μ), Studia Math.30 (1968), 231–246.
A. Pelczynski,On C(S)-subspaces of separable Banach spaces, Studia Math.31 (1968), 513–522.
A. Pelczynski,Projections in certain Banach spaces, Studia Math.19 (1960), 209–228.
H. P. Rosenthal,On injective Banach spaces and the spaces L ∞(μ)for finite measures μ, Acta Math.124 (1970), 205–248.
C. Stegall,Banach space whose duals contain l 1(Λ)with applications to the study of dual L 1(μ)spaces, submitted to Trans. Amer. Math. Soc.
Author information
Authors and Affiliations
Additional information
The research for this paper was partially supported by NSF-GP-30798X.
An erratum to this article is available at http://dx.doi.org/10.1007/BF02757135.
Rights and permissions
About this article
Cite this article
Rosenthal, H.P. On factors ofC([0, 1]) with non-separable dual. Israel J. Math. 13, 361–378 (1972). https://doi.org/10.1007/BF02762811
Issue Date:
DOI: https://doi.org/10.1007/BF02762811