Abstract
The Dieudonne-Schwartz Theorem for bounded sets in strict inductive limits does not hold for general inductive limits E = indlimEn. It does if each En C Emrn-i and all the En are Frechet spaces. A counterexample shows that this condition is not necessary. When E is a strict inductive limit of metrizable spaces E, this condition is equivalent to the condition that each bounded set in E is contained in some En. Let Ei C E2 C • • • be a sequence of locally convex spaces and
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