Abstract
The Dieudonné-Schwartz theorem for bounded sets in strict inductive limits does not hold for general inductive limits. A set B bounded in an inductive limit E = ind lim E n E = {\operatorname {ind}}\;\lim {E_n} of locally convex spaces may not be contained in any E n {E_n} . If, however, each E n {E_n} is closed in E, then B is contained in some E n {E_n} , but may not be bounded there.
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