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## Studia Mathematica

1994 | 111 | 2 | 153-162
Tytuł artykułu

### Topologies and bornologies determined by operator ideals, II

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EN
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EN
Let 𝔄 be an operator ideal on LCS's. A continuous seminorm p of a LCS X is said to be 𝔄- continuous if $Q̃_p ∈ 𝔄^{inj}(X,X̃_p)$, where $X̃_p$ is the completion of the normed space $X_p = X/p^{-1}(0)$ and $Q̃_p$ is the canonical map. p is said to be a Groth(𝔄)- seminorm if there is a continuous seminorm q of X such that p ≤ q and the canonical map $Q̃_{pq} : X̃_q → X̃_p$ belongs to $𝔄(X̃_q,X̃_p)$. It is well known that when 𝔄 is the ideal of absolutely summing (resp. precompact, weakly compact) operators, a LCS X is a nuclear (resp. Schwartz, infra-Schwartz) space if and only if every continuous seminorm p of X is 𝔄-continuous if and only if every continuous seminorm p of X is a Groth(𝔄)-seminorm. In this paper, we extend this equivalence to arbitrary operator ideals 𝔄 and discuss several aspects of these constructions which were initiated by A. Grothendieck and D. Randtke, respectively. A bornological version of the theory is also obtained.
Słowa kluczowe
EN
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
153-162
Opis fizyczny
Daty
wydano
1994
otrzymano
1993-07-19
Twórcy
autor
• Department of Applied Mathematics, National Sun Yat-sen University, Kao-hsiung, 80424, Taiwan, R.O.C., wong@math.nsysu.edu.tw
Bibliografia
• [1] L. Franco and C. Piñeiro, The injective hull of an operator ideal on locally convex spaces, Manuscripta Math. 38 (1982), 333-341.
• [2] H. Hogbe-Nlend, Bornologies and Functional Analysis, Math. Stud. 26, North-Holland, Amsterdam, 1977.
• [3] H. Hogbe-Nlend, Nuclear and Co-Nuclear Spaces, Math. Stud. 52, North-Holland, Amsterdam, 1981.
• [4] G. J. O. Jameson, Summing and Nuclear Norms in Banach Space Theory, London Math. Soc. Students Text 8, Cambridge University Press, Cambridge, 1987.
• [5] H. Jarchow, Locally Convex Spaces, Teubner, Stuttgart, 1981.
• [6] H. Jarchow, On certain locally convex topologies on Banach spaces, in: Functional Analysis: Surveys and Recent Results, III, K. D. Bierstedt and B. Fuchssteiner (eds.), North-Holland, Amsterdam, 1984, 79-93.
• [7] H. Junek, Locally Convex Spaces and Operator Ideals, Teubner-Texte Math. 56, Teubner, Leipzig, 1983.
• [8] M. Lindstorm, A characterization of Schwartz spaces, Math. Z. 198 (1988), 423-430.
• [9] A. Pietsch, Nuclear Locally Convex Spaces, Springer, Berlin, 1972.
• [10] A. Pietsch, Operator Ideals, North-Holland, Amsterdam, 1980.
• [11] D. Randtke, Characterizations of precompact maps, Schwartz spaces and nuclear spaces, Trans. Amer. Math. Soc. 165 (1972), 87-101.
• [12] H. H. Schaefer, Topological Vector Spaces, Springer, Berlin, 1971.
• [13] I. Stephani, Injektive Operatorenideale über der Gesamtheit aller Banachräume und ihre topologische Erzeugung, Studia Math. 38 (1970), 105-124.
• [14] I. Stephani, Surjektive Operatorenideale über der Gesamtheit aller Banachräume, Wiss. Z. Friedrich-Schiller-Univ. Jena, Math.-Natur. Reihe 21 (1972), 187-216.
• [15] I. Stephani, Surjektive Operatorenideale über der Gesamtheit aller Banachräume und ihre Erzeugung, Beiträge Anal. 5 (1973), 75-89.
• [16] I. Stephani, Generating system of sets and quotients of surjective operator ideals, Math. Nachr. 99 (1980), 13-27.
• [17] I. Stephani, Generating topologies and quotients of injective operator ideals, in: Banach Space Theory and Its Applications (Proc., Bucharest 1981), Lecture Notes in Math. 991, Springer, Berlin, 1983, 239-255.
• [18] N.-C. Wong and Y.-C. Wong, Bornologically surjective hull of an operator ideal on locally convex spaces, Math. Nachr. 160 (1993), 265-275.
• [19] Y.-C. Wong, The Topology of Uniform Convergence on Order-Bounded Sets, Lecture Notes in Math. 531, Springer, Berlin, 1976.
• [20] Y.-C. Wong, Schwartz Spaces, Nuclear Spaces and Tensor Products, Lecture Notes in Math. 726, Springer, Berlin, 1979.
• [21] Y.-C. Wong and N.-C. Wong, Topologies and bornologies determined by operator ideals, Math. Ann. 282 (1988), 587-614.
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