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FK Spaces in Which the Sequence of Coordinate Vectors is Bounded

  • William H. Ruckle (a1)
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The work presented in this paper was initially motivated by the following question of A. Wilansky: “Is there a smallest FK-space E in which is bounded?” Here FK-space means a complete linear metric space of real or complex sequences x = (x i ) upon which the coordinate functional x → xt are continuous for each i (see [10, p. 202]), and An FK-space need not be locally convex, and therein lies the difficulty of the problem since it is easy to see that l1 is the smallest locally convex FK-space.

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References
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1. Garling, D. J. H., On symmetric sequence spaces, Proc. London Math. Soc. 16 (1966), 85-106- 2. Symmetric bases of locally convex spaces, Studia Math. 80 (1968), 163181.
3. Jones, O. T. and Retherford, J. R., On similar bases in barrelled spaces, Proc. Amer. Math. Soc. 18 (1967), 677680.
4. Köthe, G., Topological vector spaces. I (Springer, Berlin, 1970).
5. Köthe, G. and Toeplitz, O., Lineare Ràume mit unendlich vielen Koordinaten und Ringe unendlichen Matrizen, J. Reine Agnew. Math. 171 (1934), 193226.
6. Ruckle, W., Symmetric coordinate spaces and symmetric bases, Can. J. Math. 19 (1967), 828838.
7. Ruckle, W., On perfect symmetric BK-spaces, Math. Ann. 175 (1968), 121126.
8. Ruckle, W., Topologies on sequence spaces (to appear in Pacific J. Math.).
9. Singer, J., Bases in Banach spaces. I (Springer, Berlin, 1970).
10. Wilansky, A., Functional analysis (Blaisdell, New York, 1964).
11. Gramsch, B., Die Klasse metrisher linearer Raume L(Φ), Math. Ann. 171 (1967), 6178.
12. Nakano, H., Concave modulares, J. Math. Loc. Japan 5 (1953), 2949.
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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