Köthe’s example of an incomplete LB-space

Kucera, J.; McKennon, K.

doi:10.1090/S0002-9939-1985-0766531-1

Köthe’s example of an incomplete LB-space
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by J. Kucera and K. McKennon

Proc. Amer. Math. Soc. 93 (1985), 79-80

DOI: https://doi.org/10.1090/S0002-9939-1985-0766531-1

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Abstract:

The inductive limit of reflexive Banach spaces is fast complete. But the inductive limit of reflexive Fréchet spaces may not be fast complete. Also, an example of a complete, reflexive, nuclear space with a quotient space which is not fast complete is given.

References

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Helmut H. Schaefer, Topological vector spaces, Graduate Texts in Mathematics, Vol. 3, Springer-Verlag, New York-Berlin, 1971. Third printing corrected. MR 0342978, DOI 10.1007/978-1-4684-9928-5

Closed graph theorems and webbed spaces

J. Kučera and K. McKennon, Dieudonné-Schwartz theorem on bounded sets in inductive limits, Proc. Amer. Math. Soc. 78 (1980), no. 3, 366–368. MR 553378, DOI 10.1090/S0002-9939-1980-0553378-X
Jing Hui Qiu, Some results on bounded sets in inductive limits, Acta Math. Sinica 29 (1986), no. 2, 280–284 (Chinese). MR 855717