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Continuity of operators in spaces of vector-functions; applications to the theory of bases. (Russian. English summary) Zbl 0637.46032

Let (T,\(\Sigma\),\(\mu)\) be a measure space and let X be a Banach space. The symbol E(X) denotes the space of all measurable functions \(\vec f:T\to X\) such that \(\|\| \vec f(t)\|_ X\|_{\|| t}<\infty\). The note contains new and important results connected with bases in the space E(X), where E denotes the rearrangement invariant Banach function spaces. The main results of this note are theorems 1.1 and 2.1 in which the author has investigated unconditional bases and Banach spaces with unconditional martingale property. Some results of the part 2 are very interesting in partcularly for Orlicz spaces and Lorentz spaces.
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.)
46E40 Spaces of vector- and operator-valued functions
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
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