Topologies and bornologies determined by operator ideals

Yau-Chuen Wong Ngai-Ching Wong

Functional Analysis mathscidoc:1910.43661

Mathematische Annalen, 282, (4), 587-614, 1988.12
The notion of generating topologies, introduced by Stephani El0], is useful for studying the injective hull of an operator ideal. Using Randtke's idea (see [8, p. 90] or [12,(3.2. 1) and (3.2. 7)]), we can characterize generating topologies in terms of seminorms which satisfy some expected properties (see Lemma 3.3 and Theorems 3.4 and 3.9). By a well-known and useful idea of Grothendieck, the dual notions of generating topologies and ideal-topologies, the so-called generating bornologies, are given and studied in Sect. 4. In term of ideal-bornologies, the surjective hull of an operator ideal on Banach spaces is given (see Lemma 4.8 and Theorem 4.10). In terms of these two dual concepts, we are able to classify locally convex spaces, and to study their dual results. For instance, we show that if 9.1 is a symmetric (resp. completely symmetric) operator ideal on Banach spaces then a Banach space E is an 9~-topological
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  title={Topologies and bornologies determined by operator ideals},
  author={Yau-Chuen Wong, and Ngai-Ching Wong},
  booktitle={Mathematische Annalen},
Yau-Chuen Wong, and Ngai-Ching Wong. Topologies and bornologies determined by operator ideals. 1988. Vol. 282. In Mathematische Annalen. pp.587-614.
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