Dual local completeness

Saxon, Stephen; Ruiz, L.

doi:10.1090/S0002-9939-97-03864-1

Dual local completeness
HTML articles powered by AMS MathViewer

by Stephen A. Saxon and L. M. Sánchez Ruiz PDF

Proc. Amer. Math. Soc. 125 (1997), 1063-1070 Request permission

Abstract:

The 1971 articles in which Saxon-Levin and Valdivia independently proved their Theorem feature two conditions equivalent to dual local completeness. One became Ruess’ property (LC). The other is among new characterizations previously known only as necessary conditions.

References

M. De Wilde and C. Houet, On increasing sequences of absolutely convex sets in locally convex spaces, Math. Ann. 192 (1971), 257–261. MR 288549, DOI 10.1007/BF02075355
J. C. Ferrando, M. López Pellicer and L. M. Sánchez Ruiz, Metrizable Barrelled Spaces, Pitman RNMS 332, Longman, 1995.
Taqdir Husain, Two new classes of locally convex spaces, Math. Ann. 166 (1966), 289–299. MR 212535, DOI 10.1007/BF01360784
N. J. Kalton, Some forms of the closed graph theorem, Proc. Cambridge Philos. Soc. 70 (1971), 401–408. MR 301476, DOI 10.1017/s0305004100050039
Gottfried Köthe, Topological vector spaces. I, Die Grundlehren der mathematischen Wissenschaften, Band 159, Springer-Verlag New York, Inc., New York, 1969. Translated from the German by D. J. H. Garling. MR 0248498
Michael Kunzinger, Barrelledness, Baire-like- and $(LF)$-spaces, Pitman Research Notes in Mathematics Series, vol. 298, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. MR 1267058
Pedro Pérez Carreras and José Bonet, Barrelled locally convex spaces, North-Holland Mathematics Studies, vol. 131, North-Holland Publishing Co., Amsterdam, 1987. Notas de Matemática [Mathematical Notes], 113. MR 880207
W. Roelcke, On the finest locally convex topology agreeing with a given topology on a sequence of absolutely convex sets, Math. Ann. 198 (1972), 57–80. MR 324353, DOI 10.1007/BF01420498
W. Ruess, A Grothendieck representation for the completion of cones of continuous seminorms, Math. Ann. 208 (1974), 71–90. MR 361702, DOI 10.1007/BF01432521
Wolfgang Ruess, Generalized inductive limit topologies and barrelledness properties, Pacific J. Math. 63 (1976), no. 2, 499–516. MR 415257, DOI 10.2140/pjm.1976.63.499
W. Ruess, Closed graph theorems for generalized inductive limit topologies, Math. Proc. Cambridge Philos. Soc. 82 (1977), no. 1, 67–83. MR 445268, DOI 10.1017/S030500410005369X
S. Saxon, Basis Cone Base Theory, Doctoral Dissertation, Florida State University, Tallahassee, Fla., 1969.
Stephen A. Saxon, Nuclear and product spaces, Baire-like spaces, and the strongest locally convex topology, Math. Ann. 197 (1972), 87–106. MR 305010, DOI 10.1007/BF01419586
S. A. Saxon and M. D. Levin, Every countable-codimensional subspace of a barrelled space is a barrelled space, Abstract 660-44, Notices of the Amer. Math. Soc. 15(1968), 1020.
Stephen Saxon and Mark Levin, Every countable-codimensional subspace of a barrelled space is barrelled, Proc. Amer. Math. Soc. 29 (1971), 91–96. MR 280972, DOI 10.1090/S0002-9939-1971-0280972-0
Manuel Valdivia, Absolutely convex sets in barrelled spaces, Ann. Inst. Fourier (Grenoble) 21 (1971), no. 2, 3–13 (English, with French summary). MR 333643, DOI 10.5802/aif.368
Manuel Valdivia, Mackey convergence and the closed graph theorem, Arch. Math. (Basel) 25 (1974), 649–656. MR 374856, DOI 10.1007/BF01238743
Manuel Valdivia, On quasi-completeness and sequential completeness in locally convex spaces, J. Reine Angew. Math. 276 (1975), 190–199. MR 388033, DOI 10.1515/crll.1975.276.190
Manuel Valdivia, Topics in locally convex spaces, Notas de Matemática [Mathematical Notes], vol. 85, North-Holland Publishing Co., Amsterdam-New York, 1982. MR 671092
J. H. Webb, Sequential convergence in locally convex spaces, Proc. Cambridge Philos. Soc. 64 (1968), 341–364. MR 222602, DOI 10.1017/s0305004100042900
J. H. Webb, Countable-codimensional subspaces of locally convex spaces, Proc. Edinburgh Math. Soc. (2) 18 (1972/73), 167–172. MR 318824, DOI 10.1017/S0013091500009895
Albert Wilansky, Modern methods in topological vector spaces, McGraw-Hill International Book Co., New York, 1978. MR 518316