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The author wants to thank Professors J. Bonet and S. Dierolf for making him benefit of their experience and for several helpful conversations (in particular Example 5 is due to S. Dierolf).
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Peris, A. Some results on Fréchet spaces with the density condition. Arch. Math 59, 286–293 (1992). https://doi.org/10.1007/BF01197327
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DOI: https://doi.org/10.1007/BF01197327