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Given X a Frechet Montel space, no infinite dimensional subspace of X is normable. We show that the converse implication is not true in general. In fact we provide here a Frechet space X (moreover, X is a perfect Frechet space), which is not Montel but does not contain an infinite dimensional normable subspace. Clearly, given X a Frechet Montel space, none of the infinite dimensional sub- spaces of X is normable. The converse implication is true for certain classes of Frechet spaces (e.g. echelon sequence spaces, X-K6the sequence space [2], closed subspaces of the reduced projective limits of ?p,A-spaces p > 2 [1], ...). However, we give here a counterexample in order to prove that the converse implication is not true in general. In fact, let p E R, p > 1, and let us consider the following perfect Frechet space AP = nitp= n 1p+1/n = (iP+ I/n, inm), P>p nEN Inm : IP+l/m r 1p+1/n n results of Dubinsky [3], AP is a, not Montel, perfect Frechet space. On the other hand, if AP contains an infinite dimensional normable subspace, say B, then it is easy to check that there is ko E N so that B is isomorphic to a subspace of 1p+1/k for every k > ko, but this is not possible since IP and 1r are totally incomparable spaces if p 4 r [4]. Thus Ap is a not Montel, Frechet space, without infinite dimensional normable subspaces. REFERENCES 1. J. C. Diaz, Montel subspaces in the echelon Kothe spaces (to appear). 2. S. F. Bellenot, Basic sequences in non-Schwartz-Fre'chet spaces, Trans. Amer. Math. Soc. 258 (1980), 199-216. 3. E. Dubinsky, Perfect Fre'chet spaces, Math. Ann. 174 (1967), 186-194. 4. J. Lindenstrauss and L. Tzafiri, Classical Banach spaces. I, Surveys in Math, Springer- Verlag, 1979. CATEDRA DE MATEMAJTICAS, E.T.S.I. AGR6NOMOS, UNIVERSIDAD DE CORDOBA, 14004 C6RDOBA, SPAIN Received by the editors September 6, 1985. 1980 Mathematics Subject Classification. Primary 46A14. ?1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page

An example of Fréchet space, not Montel, without infinite-dimensional normable subspaces