  A subspace $X$ of a vector space over a field $K$ is hyperinvariant with
respect to an endomorphism $f$ of $V$ if it is invariant for all endomorphisms
of $V$ that commute with $f$. We assume that $f$ is locally nilpotent, that is,
every $ x \in V $ is annihilated by some power of $f$, and that $V$ is an
infinite direct sum of $f$-cyclic subspaces. In this note we describe the
lattice of hyperinvariant subspaces of $V$. We extend results of Fillmore,
Herrero and Longstaff (Linear Algebra Appl. 17 (1977), 125--132) to infinite
dimensional spaces.
