On regularity of inductive limits.(English)Zbl 0889.46003

Let $$(E_n,\tau_n)_{n\in\mathbb{N}}$$ be a sequence of Hausdorff locally convex vector spaces with continuous injections $$E_n\subset E_{n+1}$$ and let $$(E,\tau):=\varinjlim_{n\in\mathbb{N}}E_n$$. $$E$$ is said to be $$\alpha-$$regular, if each bounded set in $$E$$ is contained in some $$E_n$$, and $$\beta-$$regular if each bounded set in $$E$$ which is contained in some $$E_n$$ is bounded in some $$E_m$$. The authors show among others that each of the following properties implies the next one (and all are equal in case the $$E_n$$ are normable):
1) Each $$E_n$$ is closed in $$E$$.
2) For every $$n\in\mathbb{N}$$ there is an $$m\geq n$$ such that $$\text{cl}_E E_n\subset E_m$$.
3) There exists a sequence $$(G_n)$$ of neighbourhoods of zero in $$E_n$$ such that for each $$n\in\mathbb{N}$$ there is an $$m\geq n$$ with $$\overline{\text{conv}}_E\bigcup_{k=0}^n G_k\subset E_m$$.
4) $$E$$ is $$\alpha-$$regular.
Moreover, $$E$$ is shown to be $$\beta-$$regular if it satisfies the following property (H): Let $$E\subset E_n$$ be a set which is bounded in $$E$$. Denote by $$E_B$$ the span of $$B$$. Then the topology on $$E_B$$ induced by $$\sigma(E_n,E'_n)$$ is contained in the topology on $$E_B$$ inherited by $$\tau$$. Moreover, the authors exhibit an example of an inductive limit where all $$E_n$$ are Hilbert spaces which fails to possess property (H).
Reviewer: C.Fenske (Gießen)

MSC:

 46A13 Spaces defined by inductive or projective limits (LB, LF, etc.) 46M40 Inductive and projective limits in functional analysis
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References:

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