Pták homomorphism theorem revisited. (English) Zbl 1050.46502

Summary: Rodrigues’ extension of the classical Pták’s homomorphism theorem to a non-necessarily locally convex setting states that a nearly semi-open mapping between a semi-B-complete space and an arbitrary topological vector space is semi-open [B. Rodrigues, J. Aust. Math. Soc., Ser. A 47, 322–333 (1989; Zbl 0708.46013)]. In this paper we study this extension and, as a consequence of the results obtained, provide an improvement of Pták’s homomorphism theorem.


46A30 Open mapping and closed graph theorems; completeness (including \(B\)-, \(B_r\)-completeness)


Zbl 0708.46013
Full Text: DOI EuDML


[1] N. Adasch, B. Ernst and D. Keim: Topological Vector Spaces. Lecture Notes in Math. 639. Springer Verlag, 1978. · Zbl 0397.46005
[2] J. Horváth: Locally Convex Spaces. Lecture Notes in Math. 331. Springer-Verlag, 1973.
[3] G. Köthe: Topological Vector Spaces. Springer-Verlag, 1969.
[4] B. Rodrigues: On Pták homomorphism theorem. J. Austral. Math. Soc. Ser. A 47 (1989), 322-333. · Zbl 0708.46013
[5] B. Rodrigues: Some new classes of topological vector spaces with closed graph theorems. Comment. Math. Univ. Carolin. 32 (1991), 287-296. · Zbl 0778.46006
[6] L. M. Sánchez Ruiz: Condiciones de tonelación en espacios vectoriales topológicos. Tesis Doctoral, Universidad de Valencia (1988).
[7] L. M. Sánchez Ruiz: On the Banach-Steinhaus theorem between topological vector spaces and locally convex spaces. Math. Japon. 36 (1991), 143-145. · Zbl 0794.46004
[8] L. M. Sánchez Ruiz: On the closed graph theorem between topological vector spaces and Fréchet spaces. Math. Japon. 36 (1991), 271-275. · Zbl 0746.46007
[9] L. M. Sánchez Ruiz: On completeness and the closed graph theorem. Math. Japon. 36 (1991), 891-894. · Zbl 0746.46009
[10] L. M. Sánchez Ruiz: Topological Vector Spaces Without Local Convexity Conditions. Functional Analysis with Current Applications in Science, Technology and Industry, M. Brokate and A. H. Siddiqi, Editors. Pitman RNMS, Longman, 1998, pp. 37-48. · Zbl 0913.46002
[11] H. H. Schaefer: Topological Vector Spaces. Springer-Verlag, 1986. · Zbl 0212.14001
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