PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 11, Pages 3279–3285 S 0002-9939(99)05185-0 Article electronically published on May 13, 1999 DENSELY HEREDITARILY HYPERCYCLIC SEQUENCES AND LARGE HYPERCYCLIC MANIFOLDS LUIS BERNAL-GONZ ´ ALEZ (Communicated by David R. Larson) Abstract. We prove in this paper that if (Tn) is a hereditarily hypercyclic sequence of continuous linear mappings between two topological vector spaces X and Y , where Y is metrizable, then there is an infinite-dimensional lin- ear submanifold M of X such that each non-zero vector of M is hypercyclic for (Tn). If, in addition, X is metrizable and separable and (Tn) is densely hereditarily hypercyclic, then M can be chosen dense. 1. Preliminaries In this paper we make the following notations and abbreviations: R = the real line, C = the complex plane, N = the set of positive integers, TVS = topological vector space, LCS = locally convex space, HC = hypercyclic, HHC = hereditarily hypercyclic, DHC = densely hypercyclic, DHHC = densely hereditarily hypercyclic. Let X and Y be two TVSs over the same field K (= R or C) and T n : X → Y (n ∈ N) a sequence of continuous linear mappings. A vector x ∈ X is said to be HC for (T n ) if the orbit {T n x : n ∈ N} is dense in Y . Denote HC ((T n )) = {x ∈ X : x is HC for (T n )}. It is easy to prove that if Y is metrizable and separable, then HC ((T n )) is a G δ subset. The sequence (T n ) is called HC whenever HC ((T n )) is not empty. Clearly, Y must be separable in this case. (T n ) is called HHC (see [An1, Section 3]) or strongly HC (see [BoS, Section 2]) whenever every subsequence (T n k ) is HC. We say that (T n ) is DHC whenever HC ((T n )) is dense in X . Finally, we say that (T n ) is DHHC whenever every subsequence (T n k ) is DHC. It is evident that if (T n ) is DHHC, then it is DHC and HHC, and if (T n ) is either DHC or HHC, then it is HC. If X = Y and T : X → X is an operator (= continuous linear selfmapping) on X , then a vector x ∈ X is said to be HC for T if and only if it is HC for the sequence (T n ) of iterates of T . T is said to be HC (HHC, DHC, DHHC) whenever the sequence (T n ) is HC (HHC, DHC, DHHC, respectively). We denote HC(T )= HC((T n )). It is well known [Kit, Theorem 4.2] (see also [Rol]) that no finite-dimensional TVS X supports a HC operator.. The problem of the existence of a HC operator on X when X is infinite-dimensional –posed by Rolewicz [Rol] in Received by the editors February 2, 1998. 1991 Mathematics Subject Classification. Primary 47B99; Secondary 46A99, 30E10, 32A07. Key words and phrases. Hypercyclic vector, linear operator, densely hereditarily hypercyclic sequence, infinite-dimensional manifold, dense manifold, metrizable topological vector space, en- tire function of subexponential type, Runge domain, infinite order linear differential operator. This research was supported in part by DGES grant #PB96–1348 and the Junta de Andaluc´ ıa. c 1999 American Mathematical Society 3279