C-Compactness in locally K-convex spaces

✍ Scribed by N. De Grande-De Kimpe

• Publisher: Elsevier Science
• Year: 1971
• Weight: 272 KB
• Volume: 74
• Category: Article
• ISSN: 1385-7258
• DOI: 10.1016/s1385-7258(71)80024-0

✦ Synopsis

In t his paper we first prove some new proper ties of locally K -convex spaces. They all are corollaries of the basic result sa ying that t he c-compact X-convex subset s of a locally X-convex space E are t he sa me in a ll the (E, E ') -admissible topologies on E. In the case X is a local field this result can be improved to K -convex compact sets. Therefore the properties stated her e are quite different from the arehimed ian case. I n t he second part we give a chain of properties of c-compactness leading t o a characte rization of the locally c-compact, locally K-convex spaces.

The follow ing notions will be used: 1. X is a non-ar ohimedian valued field with a non trivial valuation .

X is supposed to be spher ically complete. I.e . every family of closed spheres of K, t ot ally orde red by in clusion has a no n -empty intersection.