Summary 13.4.3.label The following types of locally convex spaces are barrelled:

1. (1)

   Every locally convex space with the Baire property.

2. (2)

   Every Banach space and every Fréchet space.

3. (3)

   Inductive limits of barrelled spaces.

4. (4)

   Spaces of type (LB) and (LF).

5. (5)

   The locally convex direct sum of barrelled spaces.

6. (6)

   Products of barrelled spaces.