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**Antiproximinal sets in the Banach space \(C(\omega^k;X)\).**
*(English)*
Zbl 1075.46509

From the introduction: V. Klee, in [Proc. Colloq. Convexity (Copenhagen 1965) 168–176 (1967; Zbl 0156.36303)], denoted by \(N_1\) the class of all normed spaces containing an antiproximinal closed convex set, and by \(N_2\) the class of all normed spaces containing an antiproximinal bounded convex set. The first example of a Banach space of class \(N_2\) was found by M. Edelstein and A. C. Thompson [Pac. J. Math. 40, 553–560 (1972; Zbl 0231.46029)] – the Banach space \(c_0\) contains an antiproximinal bounded symmetric closed convex body.

In [Mat. Zametki 17, 449–457 (1995; Zbl 0327.41030)] the author has shown that the Banach space \(c\) belongs to the class \(N_2\) too, and this property is shared by any Banach space of continuous functions isomorphic to \(c_0\) (see the author’s paper [Math., Rev. Anal. Numér. Théor. Approximation, Anal. Numer. Theor. Approximation 5, 127–143, (1977; Zbl 0407.46025)]).

The aim of the present paper is to extend the results of the aforementioned paper to the vectorial valued case. Similar results for the spaces \(c_0(X)\) and \(c(X)\) were obtained by the author in [Math., Rev. Anal. Numér. Théor. Approximation, Anal. Numer. Theor. Approximation 7, 141–145 (1978; Zbl 0408.41017)] and [Commentat. Math. Univ. Carol. 38, No. 2, 247–253 (1997; Zbl 0887.41029)].

In [Mat. Zametki 17, 449–457 (1995; Zbl 0327.41030)] the author has shown that the Banach space \(c\) belongs to the class \(N_2\) too, and this property is shared by any Banach space of continuous functions isomorphic to \(c_0\) (see the author’s paper [Math., Rev. Anal. Numér. Théor. Approximation, Anal. Numer. Theor. Approximation 5, 127–143, (1977; Zbl 0407.46025)]).

The aim of the present paper is to extend the results of the aforementioned paper to the vectorial valued case. Similar results for the spaces \(c_0(X)\) and \(c(X)\) were obtained by the author in [Math., Rev. Anal. Numér. Théor. Approximation, Anal. Numer. Theor. Approximation 7, 141–145 (1978; Zbl 0408.41017)] and [Commentat. Math. Univ. Carol. 38, No. 2, 247–253 (1997; Zbl 0887.41029)].

### MSC:

46E40 | Spaces of vector- and operator-valued functions |

46B20 | Geometry and structure of normed linear spaces |