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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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