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On modular spaces over a field with valuation. (English) Zbl 0629.46067

In this note X denotes a vector space over a field K with nontrivial valuation. The author considered k-quasi-convex modulars on the space X. The main result of the note says that if the modular \(\rho\) satisfies the following two conditions:
(1) \(\rho (x_ n)\to 0\) implies \(\rho (\alpha x_ n)\to 0\) for every \(\alpha\in K,\)
(2) for every \(x\in X\) there exists \(y\in X\) such that \(2\rho\) (y)\(\leq \rho (x)\) and \(2\rho\) (x-y)\(\leq \rho (x),\)
and if the valuation is nonarchimedean, then there exists a non nontrivial continuous linear functional on the modular space \(X_{\rho}\).
Reviewer: A.Waszak

MSC:

46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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References:

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