Non-archimedean nuclearity and spaces of continuous functions. (English) Zbl 0746.54004

Let \(X\) be a zero-dimensional topological Hausdorff space. Let \(K\) be a non-archimedean valued, complete field \(K\), non-trivially valued. \(C(X):=\{f: X\to K; f\text{ is continuous}\}\). Let \(\tau_ c\) be the compact-open topology on \(C(X)\), \(\tau_ p\) the topology of pointwise convergence and \(\tau_{\sigma}\) the weak topology. The authors prove:
The following are equivalent: (i) the space \((C(X),\tau_ c)\) is nuclear; (ii) every \(\tau_ c\)-bounded subset of \(C(X)\) is \(\tau_ c\)- compactoid; (iii) every compact subset of \(X\) is finite; (iv) on \(C(X)\) the topologies \(\tau_ c\) and \(\tau_ p\) coincide; (v) on \(C(X)\) the topologies \(\tau_ c\) and \(\tau_{\sigma}\) coincide; (vi) every \(\tau_ c\)-compactoid subset of \(C(X)\) is \(\tau_ c\)-compactoid; (vii) every \(\tau_ p\)-compactoid subset of \(C(X)\) is \(\tau_ c\)-compactoid; (viii) the space \((C(X),\tau_ c)\) is of countable type and in \(C(X)\) every \(\tau_ p\)-Cauchy sequence is \(\tau_ c\)-Cauchy.


54C35 Function spaces in general topology
46A99 Topological linear spaces and related structures
Full Text: DOI


[1] De Grande-De Kimpe, N., On spaces of operators between locally \(K\)-convex spaces, Proc. Kon. Ned. Akad. v. Wet., A75, 113-129 (1972) · Zbl 0235.46013
[2] De Grande-De Kimpe, N., Non-archimedean nuclearity, Groupe d’etude d’analyse ultra-metrique (1982), Institut Henri Poincare: Institut Henri Poincare Paris · Zbl 0509.46062
[3] De Grande-De Kimpe, N., The non-archimedean space \(C^∞(X)\), Compositio math., 48, 297-309 (1983) · Zbl 0509.46063
[4] Martinez-Maurica, J.; Navarro, S., \(p\)-Adic Ascoli theorems, 3, 1, 19-27 (1990), Revista Mat. Univ. Complutense de Madrid · Zbl 0718.47036
[5] Schikhof, W., Locally convex spaces over non-spherically complete fields I-II, Bull. Soc. Math. Belgique, Vol. XXXVIII, ser. B, 187-224 (1986) · Zbl 0615.46071
[6] Van der Put, M.; Van Tiel, J., Espaces nucleaires non-archimediens, Proc. Kon. Ned. Akad. v. Wet., A70, 556-561 (1967) · Zbl 0156.13201
[7] Van Tiel, J., Espaces localement \(K\)-convexes I-II-III, Proc. Kon. Ned. Akad. v. Wet., A68, 249-289 (1965) · Zbl 0133.06502
[8] Van Rooij, A. C.M., Non-archimedean functional analysis (1978), Marcel Dekker: Marcel Dekker New York · Zbl 0396.46061
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.