Dieudonné-Schwartz theorem in inductive limits of metrizable spaces

Abstract

The Dieudonné-Schwartz Theorem for bounded sets in strict inductive limits does not hold for general inductive limits E = ind lim E n E = {\operatorname {ind}}\lim {{\text {E}}_{\text {n}}} . It does if each E ¯ n E ⊂ E m ( n ) \bar E_n^E \subset {E_{m\left ( n \right )}} and all the E n {E_n} are Fréchet spaces. A counterexample shows that this condition is not necessary. When E E is a strict inductive limit of metrizable spaces E n {E_n} , this condition is equivalent to the condition that each bounded set in E E is contained in some E n {E_n} .