Pavlov, I. V.; Skorikov, A. V. 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 × Cite Format Result Cite Review PDF