Skip to main content Accessibility help

HB-subspaces and Godun sets of subspaces in Banach spaces

  • Eve Oja (a1)


Let X be a Banach space and Y its closed subspace having property U in X. We use a net (Aα) of continuous linear operators on X such that ‖ Aα ‖ ≤ 1, Aα (X) ⊂ Y for all α, and limαg(Aαy) = g(y), yY, gY* to obtain equivalent conditions for Y to be an HB-subspace, u-ideal or h-ideal of X. Some equivalent renormings of c0 and l2 are shown to provide examples of spaces X for which K(X) has property U in L(X) without being an HB-subspace. Considering a generalization of the Godun set [3], we establish some relations between Godun sets of Banach spaces and related operator spaces. This enables us to prove e.g., that if K(X) is an HB-subspace of L(X), then X is an HB-subspace of X**—the result conjectured to be true by Å. Lima [9].



Hide All
1.Casazza, P. G. and Kalton, N. J.. Notes on approximation properties in separable Banach spaces. In Geometry of Banach spaces, Proc. Conf. Strobl 1989 (eds. Müller, P. F. X. and Schachermayer, W.). London Math. Soc. Lecture Notes Series no. 158 (Cambridge University Press, 1990), pp. 4963.
2.Godefroy, G., Kalton, N. J. and Saphar, P. D.. Idéaux inconditionnels dans les espaces de Banach. C.R. Acad. Sci. Paris, 313, Sér. 1 (1991), 845849.
3.Godefroy, G., Kalton, N. J. and Saphar, P. D.. Unconditional ideals in Banach spaces. Studia Math., 104 (1993), 1359.
4.Harmand, P., Werner, D. and Werner, W.. M-ideals in Banach spaces and Banach algebras. Lecture Notes in Math. Vol. 1547 (Springer-Verlag, 1993).
5.Hennefeld, J.. M-ideals, HB-subspaces, and compact operators. Indiana Univ. Math. J., 28 (1979), 927934.
6.Johnson, J.. Remarks on Banach spaces of compact operators. J. Fund. Anal., 32 (1979), 304311.
7.Johnson, J. and Wolfe, J.. On the norm of the canonical projection of E*** onto E1. Proc. Amer. Math. Soc., 75 (1979), 5052.
8.Lima, A.. On M-ideals and best approximation. Indiana Univ. Math. J., 31 (1982), 2736.
9.Lima, A.. Uniqueness of Hahn-Banach extensions and liftings of linear dependences. Math. Scand., 53 (1983), 97113.
10.Lima, A.. Property (wM*) and the unconditional metric compact approximation property. Studia Math., 113 (1995), 249263.
11.Lima, A., Oja, E., Rao, T. S. S. R. K. and Werner, D.. Geometry of operator spaces. Michigan Math. J., 41 (1994), 473490.
12.Lindenstrauss, J. and Tzafriri, L.. On the complemented subspaces problem. Israel J. Math., 9 (1971), 263269.
13.Oja, E.. On the uniqueness of the norm preserving extension of a linear functional in the Hahn-Banach theorem. Izv. Akad. Nauk Est. SSR, 33 (1984). 424438 (in Russian).
14.Oja, E. F.. Strong uniqueness of the extension of linear continuous functionals according to the Hahn Banach theorem. Mat. Zametki, 43 (1988), 237246 (in Russian) = Math. Notes, 43 (1988), 134–139.
15.Oja, E.. Dual de 1'espace des opérateurs linéaires continus. C.R. Acad. Sci. Paris, 309, Ser. 1 (1989), 983986.
16.Oja, E. and Pôldvere, M.. On subspaces of Banach spaces where every functional has a unique norm-preserving extension. Studia Math, 117 (1996), 289306.
17.Phelps, R. R.. Uniqueness of Hahn-Banach extensions and unique best approximation. Trans Amer. Math. Soc., 95 (1960), 238255.
18.Singer, I.. Bases in Banach spaces, Vol. 2 (Springer-Verlag, 1981).
19.Sullivan, F.. Geometric properties determined by the higher duals of a Banach space. Illinois J. Math., 21 (1977), 315331.
20.Taylor, A. E.. The extension oflinear functionals. Duke Math. J., 5 (1939), 538547.
21.Yost, D.. Approximation by compact operators between C(X) spaces. J. Approx. Th., 49 (1987), 99109.
MathJax is a JavaScript display engine for mathematics. For more information see

MSC classification

HB-subspaces and Godun sets of subspaces in Banach spaces

  • Eve Oja (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed