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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 XY, the projective tensor product X ^Y, and the injective tensor product X ˇY. If X and Y are Banach spaces then in XY we may introduce e.g. the projective crossnorm || and the injective crossnorms || . The main results of this paper are the following theorems:

(1) the space 1 ˇX can be identified with the space K(c 0 ,X) (p. 383),

(2) the space 1 ^X can be identified with the space 1 (X) (p. 385); the same holds true for vector-valued functions,

(3) the space L 1 (0,1) ^X is identified with the space L X 1 (0,1) (p. 387);

(4) L 1 (0,1) ˇX 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).

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