Completion of Schwartz Convergence Vector Spaces

✍ Scribed by Mikael Lindsteöm

• Publisher: John Wiley and Sons
• Year: 1985
• Tongue: English
• Weight: 385 KB
• Volume: 122
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19851220111

✦ Synopsis

1. Definitions and introdiiction

In this note it is shown that the completion of a SCHWARTZ space E in the category of L,L, -embedded spaces is Iinearly homeomorphic with the eJf-bidual LAM E . The connection between LA,, -embedded SCHWARTZ spaces and topological SCHWARTZ spaces is also studied.

The terminology and notations in the present note are mainly the same as in f 3 . 1 0 ~ [2] and Lrwi)sva~~ar [7]. For the sake of convenience we recall same basic definitions and notationx. Let CLV denote the category of absolutely convex convergence vector spaces r~iicl continuous linec,r mappings and let K c denote the subcategory of equable, alsolutely conves convergence vector spaces [I], By L&=-Ebd ( a = e , Jf) we denote the subcategory of CLV whose objects are the LA,-embedded spaces, i.e. slmces E for which the canonical mapping j , : E -LJL,E into the hidual is a n emhedding in CLV [a]. Thereby the dual L,E ( a = e , N ) carries the local uniform convergence [8] or the canonical MARINESCU structure [GI and the second dual is endowed with local uniform convergence. The epireflective subcategory of LJ,,,-Ebd formed by all objects, which are separated locally convex topological vector spaces is denoted by LCTopV. The bornological spaces in LJa1-Ebd are characterized as the polar bornological spaces in the sense of HOGBE-NLEND [5]

endowed with NACKEY convergence for filters.