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Linear Hahn–Banach extension operators

Published online by Cambridge University Press:  20 January 2009

Brailey Sims
Affiliation:
Department of Mathematics, Statistics and Computing ScienceUniversity of New EnglandArmidaleN.S.W. 2351Australia
David Yost
Affiliation:
Department of MathematicsInstitute of Advanced StudiesAustralian National UniversityG.P.O. Box 4A.C.T. 2601Australia
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Given any subspace N of a Banach space X, there is a subspace M containing N and of the same density character as N, for which there exists a linear Hahn–Banach extension operator from M* to X*. This result was first proved by Heinrich and Mankiewicz [4, Proposition 3.4] using some of the deeper results of Model Theory. More precisely, they used the Banach space version of the Löwenheim–Skolem theorem due to Stern [11], which in turn relies on the Löwenheim–Skolem and Keisler–Shelah theorems from Model Theory. Previously Lindenstrauss [7], using a finite dimensional lemma and a compactness argument, obtained a version of this for reflexive spaces. We shall show that the same finite dimensional lemma leads directly to the general result, without any appeal to Model Theory.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 1989

References

REFERENCES

1.Fakhoury, H., Sélections linéaires associées au Théorème de Hahn–Banach, J. Funct. Anal. 11 (1972), 436452.CrossRefGoogle Scholar
2.Foguel, S. R., On a theorem of A. E. Taylor, Proc. Amer. Math. Soc. 9 (1958), 325.CrossRefGoogle Scholar
3.Giles, J. R., Gregory, D. A. and Sims, B., Geometrical implications of upper semi-continuity of the duality mapping on a Banach space, Pacific J. Math., 79 (1978), 99109.CrossRefGoogle Scholar
4.Heinrich, S. and Mankiewicz, P., Applications of ultrapowers to the uniform and Lipschitz classification of Banach spaces, Studia Math. 73 (1982), 225251.CrossRefGoogle Scholar
5.Kakutani, S., Some characterizations of Euclidean space, Japan J. Math. 16 (1939), 9397.CrossRefGoogle Scholar
6.Lima, Å., Uniqueness of Hahn–Banach extensions and liftings of linear dependences, Math. Scand. 53 (1983), 97113.CrossRefGoogle Scholar
7.Lindenstrauss, J., On nonseparable reflexive Banach spaces, Bull. Amer. Math. Soc. 72 (1966), 967970.CrossRefGoogle Scholar
8.Pełczyński, A., Linear extensions, linear averagings, and their applications to linear topological classification of spaces of continuous functions, Dissert. Math. (Rozprawy Mat.) 58 (1968).Google Scholar
9.Phelps, R. R., Uniqueness of Hahn–Banach extensions and unique best approximation, Trans. Amer. Math. Soc. 95 (1960), 238255.Google Scholar
10.Sims, B., Ultra-techniques in Banach space theory, Queen's papers in Pure and Applied Math. 60 (1982).Google Scholar
11.Stern, J., Ultrapowers and local properties of Banach spaces, Trans. Amer. Math. Soc. 240 (1978), 231252.CrossRefGoogle Scholar
12.Sullivan, F., Geometrical properties determined by the higher duals of a Banach space, Illinois J. Math. 21 (1977), 315331.CrossRefGoogle Scholar
13.Taylor, A. E., The extension of linear functionals, Duke Math. J. 5 (1939), 538547.CrossRefGoogle Scholar