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Abstract

When A and B are closed subspace of a Hubert space X, and PA, PB are the corresponding orthogonal projections, von Neumann showed that
$${\left\lfloor {\left( {I - {P_A}} \right)\left( {I - {P_B}} \right)} \right\rfloor ^n}(x) \to (I - \frac{P}{{A + B}})(x){\kern 1pt} (x \in X)$$
(a)

This result is extended by replacing X with any smooth and uniformly convex Banach space, and A and B any closed subspaces whose corresponding metric projections PA, PB are linear. Further, it is shown that (a) holds whenever X is a uniformly smooth and uniformly convex Banach space and A, B closed subspaces such that A + B is closed.

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References

  1. [1]
    B. Atlestam and F. Sullivan, Iteration with best approximation operators, Rev. Roum. Math. Pures et Appl., 21(1976),125–131.Google Scholar
  2. [2]
    R.E. Bruck and S. Reich, Nonexpansive projections and resolvents of accretive operators in Banach spaces, preprint(1977).Google Scholar
  3. [3]
    S. P. Diliberto and E.G. Straus, On the approximation of a function of several variables by the sum of functions of fewer variables, Pacific J. Math., 1(1951), 195–210.CrossRefGoogle Scholar
  4. [4]
    N. Dunford and J.T. Schwartz, Linear Operators, Part I, Inter-science, New York, 1958.Google Scholar
  5. [5]
    C. Franchetti, On the alternating approximation method, Sezione Sci., 7 (1973), 169–175.Google Scholar
  6. [6]
    M. Golomb, “Approximation by functions of fewer variables”, in On Numerical approximation (R. Langer, ed.), University of Wisconsin Press, Madison, Wisconsin, 1959.Google Scholar
  7. [7]
    I. Halperin, The product of projection operators, Acta sci. Math. (Szeged), 23 (1962), 96–99.Google Scholar
  8. [8]
    R. A. Hirschfeld, On best approximations in normed vector spaces II, Nieuw Arch. Wisk., 6(1958), 99–107.Google Scholar
  9. [9]
    R.B. Holmes, “On the continuity of the best approximation operator”, in Proc.Symp.Infinite Dimensional Topology, Annals of Math.Studies #69, Princeton Univ.Press, 1972.Google Scholar
  10. [10]
    V. Klee, On a problem of Hirschfeld, Nieuw Archief voor Wiskunde, 11 (1963), 22–26.Google Scholar
  11. [11]
    J. von Neumann, Functional Operators — Vol.11 The Geometry of Orthogonal Spaces, Annals of Math.Studies #22, Princeton Univ.Press, 1950. This is a reprint of mimeographed lecture notes first distributed in 1933.)Google Scholar
  12. [12]
    J. von Neumann, On Rings of Operators. Reduction Theory, Annals of Math., 50 (1949), 401–485.CrossRefGoogle Scholar
  13. [13]
    G. Pantelidis, Konvergente Iterationsverfahren für flach konvexe Banachräume, Rhein.-Westf.Institut für Instrumentelle Mathematik Bonn (IIM), Serie A, 23 (1968), 4–8.Google Scholar
  14. [14]
    W. J. Stiles, A Solution to Hirschfeld’s problem, Nieuw Arch. Wisk., 13(1965), 116–119.Google Scholar
  15. [15]
    F.E. Sullivan and B. Atlestam, Descent methods in smooth, rotund spaces with applications to approximation in Lp, J. Math. Anal. Appl., 48 (1974), 155–164.CrossRefGoogle Scholar
  16. [16]
    N. Wiener, On the factorization of matrices, Comment. Math. Helv., 29 (1955), 97–111.CrossRefGoogle Scholar
  17. [17]
    N. Wiener and P. Masani, The prediction theory of multivariate stochastic processes, II: The linear predictor, Acta Math., 98 (1957), 95–137.CrossRefGoogle Scholar

Copyright information

© Springer Basel AG 1979

Authors and Affiliations

  • Frank Deutsch
    • 1
    • 2
  1. 1.Pachbereich MathematikJ.W. Goethe UniversitätFrankfurt am MainGermany
  2. 2.Department of MathematicsThe Pennsylvania State UniversityUniversity ParkUSA

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