On regularity of inductive limits. (English) Zbl 0889.46003

Bosch, Carlos; Kučera, Jan

Czech. Math. J. 45, No. 1, 171-173 (1995).

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

Keywords:

inductive limit; regularity; Dieudonné-Schwartz theorem; property (H)

References:

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[4]	Kučera, J., McKennon, K.: Dieudonné-Schwartz Theorem on bounded sets in inductive limits. Proc. AMS 78 (1980), no. 3, 366-368. · Zbl 0435.46054