The Approximation Property and Locally Convex Spaces Defined by the Ideal of Approximable Operators

✍ Scribed by Esa Nelimarkka

• Publisher: John Wiley and Sons
• Year: 1982
• Tongue: English
• Weight: 407 KB
• Volume: 107
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19821070127

✦ Synopsis

The connection between the approximation property and certain classes of locally convex spaces associated with the ideal B) of approximable operators will be discussed. It mill be shown that a FRECHET MOXTEL space has the approximation property iff it is a mixed @-space, or equivalently, iff its strong dual is a @-space. Similarly, a SILYA space has the approximation property iff it is a mixed @-space or equivalently iff it is a @-space. We shall also show that <% FRECHET SCHWARTZ space with the bounded approximation property is always a @-space.

Introduction, terminology and notations. Throughout this paper E will stand for a HAUSDOEFF locally convex topological vector space over the field of real or complex nuinhers. By a neighhorhood li of E we shall mean a I)alaiiced (>onvex neighhorhood of the origin of E , and U(E) will lie a fundamental system of such neighborhoods. Given ii C'EU(E). the gauge of I?' will I)e denoted lty p c and the fartor slmce Eiker p , I)y E,: a norm on E , Is defined I)y lI~;~~zl,,=p,(x) where yc: E -E , is the quotient mapping. If I'. V c U ( E )