Ebook: Résumé de la Théorie Métrique des Produits Tensoriels Topologiques

This essay presents a metric theory of topological tensor products of Banach spaces. It is an outgrowth of the author's thesis [Mem. Amer. Math. Soc. No. 16 (1955); MR0075539 (17,763c); cf. also Ann. Inst. Fourier Grenoble 4 (1952), 73--112; 6 (1955--1956), 117--120; MR0061754 (15,879b), 1140; MR0083688 (18,746e)] which dealt with tensor products of general locally compact spaces. Most of the questions of the general theory reduce, however, to the case of Banach spaces and the limitation of generality in the present paper is welcome. Actually the principal results of the present paper concern Hilbert space H and two classical Banach spaces of (real or complex valued) functions on a locally compact space M: the space C=C0(M) of continuous functions vanishing at infinity and the space L=L1(μ) of functions integrable with respect to a positive Radon measure μ.
§ 1 (pp. 8--19) studies general ⊗-norms and § 2 (pp. 19--40) studies the ⊗-norms connected with C and L. These paragraphs, as well as the first half of the next, are preparatory; almost all the results may be found in the author's thesis and hence the proofs are usually only indicated. The great amount of space devoted to the preparatory material is justified since it makes the present paper reasonably self-contained (this is all the more important in view of the differences in notation between it and the thesis).
§ 3 (pp. 40--57) studies the ⊗-norms connected with H. The main result is given in several forms, one of which asserts that every linear mapping of norm ≤1 of H into L1(μ) can be factored to a linear mapping of norm ≤1 of H into L2(μ) and a mapping of norm ≤12π of L2(μ) into L1(μ) obtained through pointwise multiplication with a suitable f∈L2(μ). The proof of this result, as well as that of the principal theorem of the next paragraph, involves, besides the author's standard methods, a geometrical study of finite-dimensional spaces.
§ 4 (pp. 57--74) studies relations between ⊗-norms. The main result here is (author's nomenclature) the fundamental theorem of the metric theory of tensor products. It is presented in many equivalent forms, one of which is the following: Let u be a continuous Hermitian form on C×C; then there exists μ of norm ≤h∥u∥ such that u(f,f−)≤∫ff−dμ for all f∈C. Here h is a universal constant; its best value satisfies 12π≤h≤2sinh12π (the right side may be replaced by sinh12π in the real case). This result allows reduction, via decompositions, of the study of continuous linear mappings between C, L, and H spaces to a study of such mappings from one H space to another. There are also other applications, e.g., to harmonic analysis and a proof of a generalized version (with an improved constant) of a theorem of J. E. Littlewood on bilinear forms [Quart. J. Math. Oxford Ser. 1 (1930), 164--174]. The paragraph concludes with some open problems concerning the structure of general Banach spaces [cf. the following review].
[For further results contained in the present paper we refer to the reviews of previous notes (without proof): C. R. Acad. Sci. Paris 239 (1954), 577--579, 607--609; Segundo symposium sobre algunos problemas matemáticos que se están estudianto en Latino América, 1954, Centro de Cooperación Científica de la UNESCO para América Latina, Montevideo, 1954, pp. 173--177;