Hypercyclic operators are subspace hypercyclic

In this short note, we prove that for a dense set A⊂XA⊂X (XX is a Banach space) there is a non-trivial closed subspace M⊂XM⊂X such that A∩MA∩M is dense in MM. We use this result to answer a question posed in Madore and Martínez-Avendaño (2011) [9]. In particular, we show that every hypercyclic operator is subspace-hypercyclic.