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1995-1996 | 117 | 3 | 289-306
Tytuł artykułu

On subspaces of Banach spaces where every functional has a unique norm-preserving extension

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let X be a Banach space and Y a closed subspace. We obtain simple geometric characterizations of Phelps' property U for Y in X (that every continuous linear functional g ∈ Y* has a unique norm-preserving extension f ∈ X*), which do not use the dual space X*. This enables us to give an intrinsic geometric characterization of preduals of strictly convex spaces close to the Beauzamy-Maurey-Lima-Uttersrud criterion of smoothness. This also enables us to prove that the U-property of the subspace K(E,F) of compact operators from a Banach space E to a Banach space F in the corresponding space L(E,F) of all operators implies the U-property for F in F** whenever F is isomorphic to a quotient space of E.
Słowa kluczowe
Czasopismo
Rocznik
Tom
117
Numer
3
Strony
289-306
Opis fizyczny
Daty
wydano
1996
otrzymano
1995-07-12
poprawiono
1995-09-11
Twórcy
autor
  • Institute of Pure Mathematics, Tartu University, Vanemuise 46, EE-2400 Tartu, Estonia , eveoja@math.ut.ee
  • nstitute of Pure Mathematics, Tartu University, Vanemuise 46, EE-2400 Tartu, Estonia , mert@math.ut.ee
Bibliografia
  • [1] T. Andô, On some properties of convex functions, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 8 (1960), 413-418.
  • [2] B. Beauzamy, Introduction to Banach Spaces and Their Geometry, North-Holland Math. Stud. 68, North-Holland, Amsterdam, 1982.
  • [3] B. Beauzamy et B. Maurey, Points minimaux et ensembles optimaux dans les espaces de Banach, J. Funct. Anal. 24 (1977), 107-139.
  • [4] P. K. Belobrov, Minimal extension of linear functionals to second dual spaces, Mat. Zametki 27 (1980), 439-445 (in Russian).
  • [5] P. G. Casazza and N. J. Kalton, Notes on approximation properties in separable Banach spaces, in: Geometry of Banach Spaces, Proc. Conf. Strobl 1989, P. F. X. Müller and W. Schachermayer (eds.), London Math. Soc. Lecture Note Ser. 158, Cambridge Univ. Press, 1990, 49-63.
  • [6] S. R. Foguel, On a theorem by A. E. Taylor, Proc. Amer. Math. Soc. 9 (1958), 325.
  • [7] G. Godefroy, Points de Namioka, espaces normants, applications à la théorie isométrique de la dualité, Israel J. Math. 38 (1981), 209-220.
  • [8] G. Godefroy, N. J. Kalton et P. D. Saphar, Idéaux inconditionnels dans les espaces de Banach, C. R. Acad. Sci. Paris Sér. I 313 (1991), 845-849.
  • [9] G. Godefroy, N. J. Kalton et P. D. Saphar, Unconditional ideals in Banach spaces, Studia Math. 104 (1993), 13-59.
  • [10] P. Harmand, D. Werner, and W. Werner, M-Ideals in Banach Spaces and Banach Algebras, Lecture Notes in Math. 1547, Springer, Berlin, 1993.
  • [11] J. Hennefeld, M-ideals, HB-subspaces, and compact operators, Indiana Univ. Math. J. 28 (1979), 927-934.
  • [12] J. Johnson, Remarks on Banach spaces of compact operators, J. Funct. Anal. 32 (1979), 304-311.
  • [13] Å. Lima, Intersection properties of balls and subspaces in Banach spaces, Trans. Amer. Math. Soc. 227 (1977), 1-62.
  • [14] Å. Lima, M-ideals of compact operators in classical Banach spaces, Math. Scand. 44 (1979), 207-217.
  • [15] Å. Lima, Uniqueness of Hahn-Banach extensions and liftings of linear dependences, ibid. 53 (1983), 97-113.
  • [16] Å. Lima, E. Oja, T. S. S. R. K. Rao, and D. Werner, Geometry of operator spaces, Michigan Math. J. 41 (1994), 473-490.
  • [17] Å. Lima and U. Uttersrud, Centers of symmetry in finite intersections of balls in Banach spaces, Israel J. Math. 44 (1983), 189-200.
  • [18] E. Oja, On the uniqueness of the norm preserving extension of a linear functional in the Hahn-Banach theorem, Izv. Akad. Nauk Est. SSR Ser. Fiz. Mat. 33 (1984), 424-438 (in Russian).
  • [19] E. Oja, Strong uniqueness of the extension of linear continuous functionals according to the Hahn-Banach theorem, Mat. Zametki 43 (1988), 237-246 (in Russian); English transl.: Math. Notes 43 (1988), 134-139.
  • [20] E. Oja, Dual de l'espace des opérateurs linéaires continus, C. R. Acad. Sci. Paris Sér. I 309 (1989), 983-986.
  • [21] E. Oja, HB-subspaces and Godun sets of subspaces in Banach spaces, preprint, 1995.
  • [22] R. R. Phelps, Uniqueness of Hahn-Banach extensions and unique best approximation, Trans. Amer. Math. Soc. 95 (1960), 238-255.
  • [23] M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Dekker, New York, 1991.
  • [24] I. Singer, Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces, Grundlehren Math. Wiss. 171, Springer, Berlin, 1970.
  • [25] M. A. Smith and F. Sullivan, Extremely smooth Banach spaces, in: Banach Spaces of Analytic Functions, Proc. Conf. Kent, Ohio, 1976, J. Baker, C. Cleaver, and J. Diestel (eds.), Lecture Notes in Math. 604, Springer, Berlin, 1977, 125-137.
  • [26] F. Sullivan, Geometrical properties determined by the higher duals of a Banach space, Illinois J. Math. 21 (1977), 315-331.
  • [27] A. E. Taylor, The extension of linear functionals, Duke Math. J. 5 (1939), 538-547.
  • [28] S. L. Troyanski, An example of a smooth space whose dual is not strictly normed, Studia Math. 35 (1970), 305-309 (in Russian).
  • [29] L. P. Vlasov, Approximative properties of sets in normed linear spaces, Uspekhi Mat. Nauk 28 (6) (1973), 3-66 (in Russian).
  • [30] D. Werner, M-ideals and the "basic inequality", J. Approx. Theory 76 (1994), 21-30.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-smv117i3p289bwm
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