Abstract
Let E be the inductive limit of a family of locally convex topological vector spaces El. We introduce the concept of partitions of unity in E, which extends the usual partitions of unity in function spaces. Inductive limits admitting such a partition of unity have many interesting properties, similar to those of the strict inductive limits. They also have rather unusual properties concerning duals and spaces of mappings, which seem to be unknown, even in spaces likeD(Ω) andE(Ω). Some examples of such inductive limits are quoted at the end of the paper.
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