The three-space problem for 𝐿¹

Talagrand, Michel

Journal of the American Mathematical Society, Tome 03 (1990) no. 1, pp. 9-29

Résumé

We construct a subspace $X$ of ${L^1}$ such that $X$ is an ${l^1}$-sum of spaces isomorphic to ${l^1}$ but such that ${L^1}/X$ does not contain a copy of ${L^1}$. We also construct two Banach spaces ${E_1}$, ${E_2}$ that do not contain a copy of ${L^1}$ but such that ${E_1} \times {E_2}$ contains a copy of ${L^1}$. Moreover, the projections of ${L^1}$ on each factor are one-to-one, and the images of the unit ball of ${L^1}$ are closed. These examples settle questions of J. Lindenstrauss, P. Pelczynski, J. Bourgain, and H. P. Rosenthal.

Détail

DOI : 10.1090/S0894-0347-1990-1013926-7

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