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On non-solid Köthe space of \(\Delta_ 2\) type. (English) Zbl 0708.46034
In the Köthe space \((L,\| \cdot \|_ L)\) the author defines the functional \[ H(f)=\inf \{\sum^{n}_{i=1}\| \chi_{A_ i}f\|_ L:\;\| \chi_{A_ i}f\|_ L\leq 1,\quad i=1,2,...,n\}, \] where the infimum is taken over all partitions \(\{A_ 1,...,A_ n\}\). By means of the functional H one may introduce the concept of (\(\Delta\) \({}_ 2)\) condition to spaces of this type. The main results of the note are interesting properties of the functional H (part 2) and applications of the functional H and the \((\Delta_ 2)\) condition to characterize Köthe spaces (part 3).
Reviewer: A.Waszak
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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