## Minimal-Hausdorff $$p$$-adic locally convex spaces.(English)Zbl 0828.46068

Summary: We characterize, in various ways, those Hausdorff locally convex spaces over a non-archimedean valued field $$K$$ that do not admit a strictly weaker Hausdorff locally convex topology. Our results extend the ones obtained by N. De Grande-De Kimpe in [Indag. Math. 33, 176-180 (1971; Zbl 0209.144)], for spherically complete $$K$$. For an analogous theory for compactoids instead of locally convex spaces we refer to 2nd Conf. on $$p$$-adic Functional Analysis, Santiago de Chile (1992).

### MSC:

 46S10 Functional analysis over fields other than $$\mathbb{R}$$ or $$\mathbb{C}$$ or the quaternions; non-Archimedean functional analysis 46A03 General theory of locally convex spaces

Zbl 0209.144
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### References:

 [1] De Grande-De Kimpe, N., c-compactness in locally K-convex spaces, Indag. Math.33 (1971), 176-180. · Zbl 0209.14404 [2] van Rooij, A.C.M., Non-Archimedean Functional Analysis. Marcel Dekker, New York (1978). · Zbl 0396.46061 [3] Schikhof, W.H., Locally convex spaces over non-spherically complete valued fields, Bull. Soc. Math. Belg.XXXVIII, serie B (1986), 187-224. · Zbl 0615.46071 [4] Schikhof, W.H., p-adic local compactoids, Report 8802, Department of Mathematics, Catholic University, Nijmegen, The Netherlands (1988). · Zbl 0712.46045 [5] Schikhof, W.H., More on duality between p-adic Banach spaces and compactoids, Report 9301, Department of Mathematics, Catholic University, Nijmegen, The Netherlands (1993). [6] Schikhof, W.H., The equalization of p-adic Banach spaces and compactoids. To appear in the Proceedings of the Second International Conference on p-adic Functional Analysis, held in Santiago, Chile, 1992. [7] van Tiel, J., Espaces localement K-convexes. Indag. Math.27 (1965), 249-289. · Zbl 0133.06502
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