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A theorem of Littlewood, Orlicz, and Grothendieck about sums in \(L^1(0,1)\). (English) Zbl 0974.46031
In this very valuable paper for given two linear spaces \(X\) and \(Y\) we consider the space \(X\otimes Y\), the projective tensor product \(X\widehat\otimes Y\), and the injective tensor product \(X\check\otimes Y\). If \(X\) and \(Y\) are Banach spaces then in \(X\otimes Y\) we may introduce e.g. the projective crossnorm \(||_\wedge\) and the injective crossnorms \(||_\vee\). The main results of this paper are the following theorems:
(1) the space \(\ell^1\check\otimes X\) can be identified with the space \(K(c_0, X)\) (p. 383),
(2) the space \(\ell^1\widehat\otimes X\) can be identified with the space \(\ell^1(X)\) (p. 385); the same holds true for vector-valued functions,
(3) the space \(L^1(0,1)\widehat\otimes X\) is identified with the space \(L^1_X(0,1)\) (p. 387);
(4) \(L^1(0, 1)\check\otimes X\) is isometrically isomorphic to the completion of the space \(P_X(0, 1)\) (p. 389).
Very interesting and valuable are comments and remarks connected with the theorem of Grothendieck (p. 392) and the theorem of Littlewood-Orlicz-Grothendieck (p. 393).

MSC:
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
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