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The space \(C_ 0(p)\) over valued fields. (English) Zbl 0603.46069

The authors while studying the space \(c_ 0(p)\) in the context of i) weak and strong convergence, ii) nuclearity and iii) Schwartz space, have exhibited differences in the concept of nuclear space, Schwartz space over non-archimedean valued fields as against such spaces over the complex field. It may be noted that the concept of Schwartz space is different from that of nuclear space [as defined in M. van der Put and J. van Tiel, Indag. Math. 29, 556-561 (1967; Zbl 0156.132) without involving the concept of nuclear map]. It is also worthwhile to note that the definition of a nuclear operator according to J.-P. Serre [Inst. Hautes Etud. Sci., Publ. Math. 12, 69-85 (1962; Zbl 0104.336)] is the same as completely continuous operator (equivalently compact operators) in the non-archimedean setting. The remarks 3.4 ii), iii) which clearly state the differences are made with a view that the earlier literature may mislead future workers.

MSC:

46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
46A13 Spaces defined by inductive or projective limits (LB, LF, etc.)
46A11 Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.)
46A45 Sequence spaces (including Köthe sequence spaces)
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