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Regularity of conservative inductive limits. (English) Zbl 0968.46001
Let $$E_1\subset E_2 \subset \cdots$$ be a sequence of Hausdorff locally convex spaces equipped with topologies $$\tau_n$$. The author considers a set of conditions under which the inductive limit $$\text{ind}E_n$$ is a regular space. Let us call that $$\text{ind}E_n$$ is quasi-regular, if any bounded set in $$\text{ind}E_n$$ is bounded in some space $$E_n$$, and call conservative if for any linear subspace $$F$$ of $$\text{ind}E_n$$, it holds $$\text{ind}(F\cap E_n,\tau_n)=(F, \tau)$$. The results are
Theorem. Any sequentially complete conservative $$\text{ind} E_n$$ is quasi regular and if each space $$E_n$$ is closed in $$\text{ind}E_n$$ then $$\text{ind}E_n$$ is regular.
##### MSC:
 46A13 Spaces defined by inductive or projective limits (LB, LF, etc.) 46A30 Open mapping and closed graph theorems; completeness (including $$B$$-, $$B_r$$-completeness)
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