Kąkol, Jerzy Remarks on spherical completeness of non-archimedean valued fields. (English) Zbl 0826.46072 Indag. Math., New Ser. 5, No. 3, 321-323 (1994). Let \(K\) be a non-Archimedean valued field. J. von Tiel proved in [Indag. Math. 27, 249-289 (1965; Zbl 0133.065)]that if \(K\) is spherically complete then every locally convex space \((E, \tau)\) over \(K\) admits the Mackey topology (i.e. the finest locally convex topology \(\mu\) for which \((E, \tau)'= (E, \mu)'\)). In this note the following converse is proved. If for every local convex space over \(K\) the Mackey topology exists then \(K\) is spherically complete. The proof is elegant and ingenious. Furthermore, it is shown that each one of the related statements ‘the completion of a dual-separating \(K\)-normed space is separating’ and ‘every closed subspace of a dual-separating \(K\)-normed space is weakly closed’ is equivalent to spherical completeness of \(K\). Reviewer: W.Schikhof (Nijmegen) Cited in 3 Documents MSC: 46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis 46A20 Duality theory for topological vector spaces 46A03 General theory of locally convex spaces Keywords:spherical completeness of non-Archimedean valued fields; spherically complete; Mackey topology Citations:Zbl 0133.065 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Dierolf, S., A note on the lifting of linear and locally convex topologies on a quotient space, Collectanea Math., 31, 193-198 (1980) · Zbl 0453.46002 [2] Ingleton, A. W., The Hahn-Banach Theorem for non-archimedean valued fields, Proc. Cambridge Phil. Soc., 48, 41-45 (1952) · Zbl 0046.12001 [3] Kakol, J., The Mackey-Arens theorem for non-locally convex spaces, Collectanea Math., 41, 129-132 (1990) · Zbl 0745.46009 [4] Peck, N. T.; Porta, H., Linear topologies which are suprema of dual-less topologies, Studia Math., 57 (1973) · Zbl 0255.46009 [5] Rooij, A. C.M.van, Non-Archimedean Functional Analysis (1978), Marcel Dekker: Marcel Dekker New York · Zbl 0396.46061 [6] Schikhof, W. H., Locally convex spaces over non-spherically complete valued fields, I-II. Bull. Soc. Math. Belgique, 38, 187-224 (1986) · Zbl 0615.46071 [7] Tiel, I.van, Espaces localement \(K\)-convexes, I-III. Indag. Math., 27, 273-289 (1965) · Zbl 0133.06502 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.