The duality theorem referred to in the title is the following well-known (cf. Buck [1]) consequence of the Hahn—Banach theorem:

Theorem A. Let V be a subspace of the normed linear space X. For each f∈X
$$ \mathop {\inf }\limits_{v \in V} \left\| {f - v} \right\| = \mathop {\max }\limits_{T \in A} \left| {Tf} \right| $$
where A is the set of all linear functionals on X of norm 1 satisfying TV=0.


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  1. [1]
    R. C. Buck, Applications of duality in approximation theory. In: Approximation of Functions, ed. by H. L. Garabedian, pp. 27–42. Elsevier, Amsterdam 1965.Google Scholar
  2. [2]
    S. Karlin, Total Positivity, Vol. 1. Stanford University, Stanford, California, 1968.Google Scholar
  3. [3]
    G. Meinardus, Approximation of Functions: Theory and Numerical Methods. Springer, Berlin 1967.CrossRefGoogle Scholar
  4. [4]
    T. J. Rivlin and H. S. Shapiro, A unified approach to certain problems of approximation and minimization), J. Soc. Indust. Appi. Math., 9 (1961), 670–699.CrossRefGoogle Scholar
  5. [5]
    M. Zedek, On approximation by solutions of ordinary linear differential equations. SIAM J. Numer. Anal. 3 (1966), 360–365.CrossRefGoogle Scholar

Copyright information

© Springer Basel AG 1969

Authors and Affiliations

  • T. J. Rivlin
    • 1
  1. 1.IBM Watson Research CenterYorktown HeightsNew YorkUSA

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