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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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