A Characterization of the Quasi-Normable Fréchet Spaces

✍ Scribed by Reinhold Meise; Dietmar Vogt

• Publisher: John Wiley and Sons
• Year: 1985
• Tongue: English
• Weight: 466 KB
• Volume: 122
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19851220114

✦ Synopsis

Preface.

The class of quasi-normable locally convex spaces has been introduced by GROTHENDIECK [4]. Recently VALDIVIA [7] and BIERSTEDT, NEISE and SUMXERS [2], [3] independently gave a characterization of the quasirnormability of the FR~CRET-KOTHE spaces A(A) resp. P ( I , A ) in terms of the KOTHE matrix A . It turned out that their necessary and sufficient condition holds if and only if P(1, A ) satisfies a linear topological invariant of type (L?,) which had been introduced by VOGT and WAGSER [ 121 t o show that there is no quotient-universal nuclear FRE-CHET space.

This observation was the starting point for the present article, in which we give several characterizations of the quasi-normability of F ~C H E T spaces. The most interesting might be that a FRI~CHET space E is quasi-normable if and only if E is a quotient space of l l ( l ) @ x R ( ~l = A ( J , A), where I is an appropriate set and where R(A) is a nuclear FRI~CHET space. If E is separable, I can be replaced by N.