Some applications of projective tensor products to holomorphy

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This informative paper surveys applications of results in the theory of topological tensor products of Fréchet or (DF)-spaces to the topological structure of spaces of n-homogeneous continuous polynomials. Recent progress about tensor products which is relevant in the present survey article was obtained in the late 1980's and the 1990's by, among others, Taskinen, Defant, Díaz, Peris, Mangino, Bierstedt and the reviewer. Precise references are given in the article. R. Ryan observed in his thesis in 1980 that the space of n-homogeneous continuous polynomials on a complex locally convex space E is isomorphic to the topological dual of the complete n-fold symmetric projective tensor product of E. This duality permits one to consider four different natural topologies on the space of polynomials: the compact open topology, the topology of the uniform convergence on the bounded subsets of E, the strong topology with respect to the aforementioned duality, and the Nachbin ported topology. Some of the theorems presented in the paper have consequences for spaces of holomorphic functions defined on balanced open domains of a complex locally convex space E. Several topics are discussed in the article. An example due to J. M. Ansemil and J. Taskinen [Arch. Math. (Basel) 54 (1990), no. 1, 61–64;] of a Fréchet-Montel space E such that the compact open topology is different from the Nachbin ported topology on H(U) for every balanced open subset U of E is presented in Section 2. Section 3 includes examples due to Peris and the reviewer and to J. M. Ansemil, F. Blasco and S. Ponte [J. Math. Anal. Appl. 213 (1997), no. 2, 534–539; MR1470868 (99d:46068)] about quasinormable spaces of polynomials. In Section 4 the authors report about nice theorems due to Blasco [Arch. Math. (Basel) 70 (1998), no. 2, 147–152;] concerning barrelled spaces of polynomials defined on Köthe echelon spaces. Polynomials on stable spaces (including results due to Díaz and Dineen, and to Ansemil and Floret), and the three-space problem for the coincidence of topologies in spaces of polynomials are also considered. In the last section a recent example due to Ansemil, Blasco and Ponte [Ann. Acad. Sci. Fenn. Math. 25 (2000), no. 2, 307–316;] of a Fréchet space E such that the topology of uniform convergence on the bounded sets and the Nachbin ported topology coincide on the space of 2-homogeneous polynomials but not on the space of 3-homogeneous polynomials on E is mentioned.
Monográfico: Perspectivas en Análisis Matemático
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