Bounded sets in fast complete inductive limits

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.

Keywords

• inductive limit of locally convex spaces
• fast complete space
• webbed space
• bounded set.