Based on a couple of references, it seems that the answer is yes, see for example
[Boneta-Dierolf, 1992][1] and [Bierstedt-Bonet, 1989][2].

However, from a comment to the answer of [this MO question][3], I infer that there is a counterexample.


  [1]: http://www.sciencedirect.com/science/article/pii/S0304020808703207
  [2]: http://dmle.cindoc.csic.es/revistas/detalle.php?numero=592
  [3]: http://mathoverflow.net/questions/8443/barrelled-bornological-ultrabornological-semi-reflexive-how-are-these-us