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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:
46E30Spaces of measurable functions
46B15Summability and bases in normed spaces
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References:
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