# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 46B15 Summability and bases in normed spaces
Full Text:
##### References:
 [1] Belanger, A.; Dowling, P. N.: Two remarks on absolutely summing operators. Math. nachr. 136, 229-232 (1988) · Zbl 0654.47009 [2] Diestel, J.: An elementary characterization of absolutely summing operators. Math. ann. 196, 101-105 (1972) · Zbl 0221.46040 [3] Diestel, J.; Jarchow, H.; Tonge, A.: Absolutely summing operators. Cambridge studies in advanced mathematics 43 (1995) · Zbl 0855.47016 [4] Diestel, J.; Jr., J. J. Uhl: Vector measures. Amer. math. Soc. surveys 15 (1977) [5] Dubinsky, E.; Pelczynski, A.; Rosenthal, H. P.: On Banach spaces X for which ${\Pi}2$(L\infty,X)=$B(L\infty,X)$. Studia math. 44, 617-648 (1972) · Zbl 0262.46018 [6] Dvoretzky, A.; Rogers, C. A.: Absolute and unconditional convergence in normed linear spaces. Proc. nat. Acad. sci. USA 36, 192-197 (1950) · Zbl 0036.36303 [7] De La Madrid, J. Gil: Measures and tensors, I. Trans. amer. Math. soc. 114, 98-121 (1965) · Zbl 0186.46604 [8] Grothendieck, A.: Résumé de la théorie métrique des produits tensoriels topologiques. Bol. soc. Mat. são paulo 8, 1-79 (1953/56) [9] Hoffmann-Jørgensen, J.: Sums of independent Banach space valued random variables. Studia math. 52, 159-186 (1974) · Zbl 0265.60005 [10] Kwapien, S.: Isomorphic characterization of inner product spaces by orthogonal series with vector-valued coefficients. Studia math. 44, 583-595 (1972) · Zbl 0256.46024 [11] Kwapien, S.: On Banach spaces containing co: A supplement to the paper by J. Hoffmann-jørgensen ”sums of independent Banach space valued random variables”. Studia math. 52, 187-188 (1974) · Zbl 0295.60003 [12] Latała, R.; Oleszkiewicz, K.: On the best constant in the Khinchin--kahane inequality. Studia math. 109, 101-104 (1994) · Zbl 0812.60010 [13] Littlewood, J. E.: On bounded bilinear forms in an infinite number of variables. Quart. J. Math. (Oxford) 1, 164-174 (1930) · Zbl 56.0335.01 [14] Macphail, M. S.: Absolute and unconditional convergence. Bull. amer. Math. soc. 53, 121-123 (1947) · Zbl 0032.35702 [15] Maurey, B.; Pisier, G.: Séries de variables aléatoires vectorielles indépendantes et properiétés géometriques des espaces de Banach. Studia math. 58, 45-90 (1976) · Zbl 0344.47014 [16] Orlicz, W.: Über unbedingte konvergenz in funktionenräumem (I). Studia math. 4, 33-37 (1933) · Zbl 0008.31501 [17] Paley, R. E. A.C.; Zygmund, A.: On some series of functions I. Math. proc. Cambridge philos. Soc. 26, 337-357 (1930) · Zbl 56.0254.01 [18] Szarek, S. J.: On the best constants in the Khinchin inequality. Studia math. 58, 197-208 (1976) · Zbl 0424.42014 [19] Zygmund, A.: Trigonometric series. (1988) · Zbl 0628.42001