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The spaces \(L_{\bar p}\) with mixed norm in an infinite-dimensional torus. (English. Russian original) Zbl 0628.46021

Sov. Math. 30, No. 2, 101-104 (1986); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1986, No. 2(285), 69-72 (1986).
Let \(\bar p=(p_ 1,p_ 2,...)\) and \(T^{\infty}\) the torus which is an infinite Cartesian product of the unit circumferences. The space \(L_{\bar p}\) is provided with a mixed norm (defined as the supremum of norms involving n-dimensional torus and \((p_ 1,p_ 2,...,p_ n)\) for all n). Then the authors state without proof the conditions in terms of \(\bar p\) so that
(a) \(f_ n\downarrow 0\) implies \(\| f_ n\|_{\bar p}\downarrow 0;\)
(b) \(L_{\bar p}\) is uniformly convex; and
(c) other properties hold.
See, also, A. Benedek and R. Panzone, Duke Math. J. 28, 301- 324 (1961; Zbl 0107.089).
Reviewer: Lee Peng-Yee

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Keywords:

torus; mixed norm

Citations:

Zbl 0107.089