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

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

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

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

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

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

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Dunham
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Is a Fréchet Montel space distinguished?

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

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