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Linear functionals on some non-locally convex generalized Orlicz spaces. (English) Zbl 0647.46031
The note contains very general and important theorems on nonexistence and existence of nonzero continuous linear functionals on non-locally convex generalized Orlicz spaces of functions with values in a p-normable space. In course of these investigations the authors consider the following important condition: there exist \(z\in X\setminus \{0\}\) and a set A of positive measure such that \(\liminf_{u\to \infty}(1/u)\Phi(uz,t)>0\) for \(t\in A\) (where \(\Phi\) denotes the function defined in 1.2 of the note). All proofs are clear and it is a neat piece of work. In the last part of the note there are given valuable examples and corollaries.
Reviewer: A.Waszak
MSC:
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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