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Zbl 0974.46031
Diestel, Joe; Fourie, Jan; Swart, Johan
A theorem of Littlewood, Orlicz, and Grothendieck about sums in $L^1(0,1)$. (English)
[J] J. Math. Anal. Appl. 251, No.1, 376-394 (2000). ISSN 0022-247X

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:\par (1) the space $\ell^1\check\otimes X$ can be identified with the space $K(c_0, X)$ (p. 383),\par (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,\par (3) the space $L^1(0,1)\widehat\otimes X$ is identified with the space $L^1_X(0,1)$ (p. 387);\par (4) $L^1(0, 1)\check\otimes X$ is isometrically isomorphic to the completion of the space $P_X(0, 1)$ (p. 389).\par 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).
[Aleksander Waszak (Poznań)]

MSC 2000:: *46E30 Spaces of measurable functions
46B15 Summability and bases in normed spaces

Keywords: projective tensor product; injective tensor product; projective crossnorm; injective crossnorms; Grothendieck; theorem of Littlewood-Orlicz-Grothendieck

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Zentralblatt MATH Berlin [Germany]