International Journal of Mathematics and Mathematical Sciences 
Volume 7 (1984), Issue 3, Pages 615-617
doi:10.1155/S016117128400065X
Research notes

Bounded sets in fast complete inductive limits

Jan Kucera and Carlos Bosch

 Department of Mathematics, Washington State University, Pullman 99164, Washington, USA
Received 26 April 1984

Abstract

Let E1⊂E2⊂… be a sequence of locally convex spaces with all identity maps: En→En+1 continuous and E=indlim En fast complete. Then each set bounded in E is also bounded in some En iff for any Banach disk B bounded in E and n∈N, the closure of B⋂En in B is bounded in some Em. This holds, in particular, if all spaces En are webbed.