ABSTRACT. The Bohnenblust-Hille inequality says that the ℓ 2m m+1 -norm of the coefficients of an m-homogeneous polynomial P on C n is bounded by P ∞ times a constant independent of n, where · ∞ denotes the supremum norm on the polydisc D n . The main result of this paper is that this inequality is hypercontractive, i.e., the constant can be taken to be C m for some C > 1. Combining this improved version of the Bohnenblust-Hille inequality with other results, we obtain the following: The Bohr radius for the polydisc D n behaves asymptotically as (log n)/n modulo a factor bounded away from 0 and infinity, and the Sidon constant for the set of frequencies log n : n a positive integer ≤ N is √ N exp{(−1/ √ 2 + o(1)) √ log N log log N } as N → ∞.

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Solutions are provided to several questions concerning topologically transitive and hypercyclic continuous linear operators on Hausdorff locally convex spaces that are not Fréchet spaces. Among others, the following results are presented. (1) There exist transitive operators on the space ϕ of all finite sequences endowed with the finest locally convex topology (it was already known that there is no hypercyclic operator on ϕ). (2) The space of all test functions for distributions, which is also a complete direct sum of Fréchet spaces, admits hypercyclic operators. (3) Every separable infinite-dimensional Fréchet space contains a dense hyperplane that admits no transitive operator.

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