Advertisement

Quasi-interior Points and the Extension of Linear Functionals

  • R. E. Fullerton
  • C. C. Braunschweiger

Abstract

Let X be a real normed space, M a linear subspace of X, and X’ and M’ the spaces conjugate to X and M, respectively. It is known from the Hahn-Banach theorem [2] that to each f in M’ there corresponds at least one x’ in X’ with the same norm such that x’ (x) = f(x) for each x in M. The question of when this norm preserving extension is unique has received considerable attention. Taylor [15] and Foguel [4] have shown that this extension is unique for every subspace M of X and for each f in M’ if and only if the unit ball in X’ is strictly convex. Phelps [12] has shown that each f in M’ has a unique norm preserving extension in X’ if and only if the annihilator M┴ of M has the Haar property in X’; that is, if and only if to each x’ in X’ corresponds a unique y’ in M┴ such that ||x’, − y’|| = inf {||x’ − z’||: z’ ϵ M ┴}.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literature

  1. [1]
    Braunschweiger, C. C.: A geometric construction of L-spaces. Duke Math. J., 23 (2), 271–280 (1956).Google Scholar
  2. [2]
    Dunford, N., and J. T. Schwartz: Linear Operators I. New York: Interscience Pub. Inc. 1957.Google Scholar
  3. [3]
    Eilenberg, S.: Banach space methods in topology. Ann. Math. 43 (2) 568–579 (1942).CrossRefGoogle Scholar
  4. [4]
    Foguel, S.: On a theorem by A. E. Taylor. Proc. Amer. Math. Soc. 9, 325 (1958).CrossRefGoogle Scholar
  5. [5]
    Fullerton, R. E.: A characterization of L-spaces. Fundamenta Math. 38, 127–136 (1951).Google Scholar
  6. [6]
    Fullerton, R. E.: Quasi-interior points of cones in a linear space (unpublished research report) (1957).Google Scholar
  7. [7]
    Fullerton, R. E.: Extensions of linear functionals (unpublished research report) (1963).Google Scholar
  8. [8]
    Fullerton, R. E., and C. C. Braunschweiger: Quasi-interior points of cones. Univ. of Del. Dept. of Math. Tech. Report No. 2 (1963).Google Scholar
  9. [9]
    James, R. C.: Characterizations of reflexivity. Studia Math. 23, 205–216 (1964).Google Scholar
  10. [10]
    Leader, S.: Separation and approximation in topological vector lattices. Canad. J. Math. 11, 286–296 (1959).CrossRefGoogle Scholar
  11. [11]
    Moore, C.: Convex sets in linear topological spaces. Masters Thesis Univ. of Del. (1964).Google Scholar
  12. [12]
    Phelps, R.: Uniqueness of Hahn-Banach extensions and unique approximations. Trans. Amer. Math. Soc. 95, 238–255 (1960).Google Scholar
  13. [13]
    Schaefer, H. H.: Some spectral properties of positive linear operators. Pacific J. Math. 10, 1009–1019 (1960).Google Scholar
  14. [14]
    Schaefer, H. H.: Halbgeordnete lokalkonvexe Vektorräume III. Math. Ann. 141 (2), 113–142 (1960).CrossRefGoogle Scholar
  15. [15]
    Taylor, A. E.: The extension of Hnear functionals. Duke Math. J. 5, 538–547 (1939).CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1966

Authors and Affiliations

  • R. E. Fullerton
    • 1
  • C. C. Braunschweiger
    • 1
  1. 1.NewarkUSA

Personalised recommendations