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The Dieudonné-Schwartz theorem for $$p$$-adic inductive limits. (English) Zbl 1130.46048
A classical result of J. Dieudonné and L. Schwartz [Ann. Inst. Fourier 1, 61–101 (1949; Zbl 0035.35501)] says that, for any bounded subset $$D$$ of a strict inductive limit $$E$$ of an increasing sequence $$(E_n)$$ of $$F$$-subspaces, there exists $$n$$ such that $$D\subset E_n$$ and $$D$$ is bounded in $$E_n$$. The present paper is concerned with the validity of this result in $$p$$-adic inductive limits of locally convex spaces, as well as of some of its extensions proved in Archimedean functional analysis (i.e., over $$\mathbb R$$ or $$\mathbb C$$) since the publication of the original result. This property is called regularity and the authors study also a weaker one, called $$\alpha$$-regularity, asserting only the existence of an $$n$$ such that $$D\subset E_n$$ (without boundedness).
The theory of inductive limits of $$p$$-adic locally convex spaces was developed in two previous papers by N. De Grande-De Kimpe, J. Kakol, C. Perez-Garcia and W. H. Schikhof [Lect. Notes Pure Appl. Math. 192, 159–222 (1997; Zbl 0889.46063)] and N. De Grande-De Kimpe and A. Yu. Khrennikov [Bull. Belg. Math. Soc. - Simon Stevin 3, No. 2, 225–237 (1996; Zbl 0845.46047)]. It turns out that this result is still valid in $$p$$-adic strict LF-spaces (Corollary 3.3), but the validity of some of the extensions requires some supplementary assumptions. The authors show by examples that without these assumptions the results fail and also that the converses of some positive results do not hold.
MSC:
 46S10 Functional analysis over fields other than $$\mathbb{R}$$ or $$\mathbb{C}$$ or the quaternions; non-Archimedean functional analysis 46A13 Spaces defined by inductive or projective limits (LB, LF, etc.)
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