Arias de Reyna, J. Dense hyperplanes of first category. (English) Zbl 0419.46015 Math. Ann. 249, 111-114 (1980). Summary: In 1966 V. Klee and A. Wilansky [Research problems #13, Bull. Am. Math. Soc. 72, 656 (1966)] raised the following question: Must an hyperplane of a Banach space be either closed or of second category? This question has been studied by different methods obtaining related results. See for example A. R. Todd and S. A. Saxon [Math. Ann. 206, 23–34 (1973; Zbl 0247.46002)] and J. P. R. Christensen [Topology and Borel structure. Amsterdam etc.: North-Holland (1974; Zbl 0273.28001)].In this paper the question is solved for a separable Banach space and assuming Martin’s axiom by the following two theorems: Theorem 1. In every infinite dimensional separable Banach space there exist dense hyperplanes of first category. Theorem 2. In every infinite dimensional separable Banach space there exists a set \(P\) of dense hyperplanes such that \(\text{card}(P)=2^{(2\aleph_0)}\) and \(\cup_{H\in P} H\) is of first category. Reviewer: J. Arias de Reyna Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 5 Documents MSC: 46B10 Duality and reflexivity in normed linear and Banach spaces 54E52 Baire category, Baire spaces Keywords:hyperplane of a Banach space; second category; Martin’s axiom; separable infinite-dimensional Banach space; first category Citations:Zbl 0247.46002; Zbl 0273.28001 × Cite Format Result Cite Review PDF Full Text: DOI EuDML References: [1] Christensen, J.P.R.: Topology and Borel structure. Notas de Matematica. Amsterdam: North-Holland 1974 · Zbl 0273.28001 [2] Klee, V., Wilansky, A.: Research problems no. 13. Bull. Amer. Math. Soc.72, 656-656 (1966) · doi:10.1090/S0002-9904-1966-11547-1 [3] Lacey, E.: Separable quotients of Banach spaces. An. Acad. Brasil. Ci.44, 185-189 (1972) · Zbl 0259.46017 [4] Munroe, M.E.: Measure and integration, 2. ed. Reading: Addison-Wesley 1971 · Zbl 0227.28001 [5] Rosenthal, H.P.: Quasicomplemented subspaces of Banach spaces. J. Functional Analysis4, 176-214 (1969) · Zbl 0185.20303 · doi:10.1016/0022-1236(69)90011-1 [6] Schoenfield, J.R.: Martin’s axiom. Amer. Math. Monthly82, 610-617 (1975) · Zbl 0314.02069 · doi:10.2307/2319691 [7] Todd, A.R., Saxon, S.A.: A property of locally convex Baire spaces. Math. Ann.206, 23-34 (1973) · doi:10.1007/BF01431526 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.