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Closed graph theorems for bornological spaces. (English) Zbl 1376.46005

For bornological vector spaces over the field \(K\), where \(K=\mathbb{R}\) or \(K=\mathbb{C}\), F. Gach [Acta Math. Univ. Comen., New Ser. 75, No. 2, 209–218 (2006; Zbl 1164.46301)] has presented a theorem that includes two classical results: M. De Wilde’s [Bull. Soc. R. Sci. Liège 40, 116–118 (1971; Zbl 0216.40603)] closed graph theorem for webbed locally convex spaces, and a bornological closed graph theorem of N. Popa [C. R. Acad. Sci., Paris, Sér. A273, 294–297 (1971; Zbl 0217.44401)]. In the paper under review, the theorem of Gach is extended to the non-archimedean setting.
The author first gives an introductory overview of bornological vector spaces over a complete non-trivially valued field \(K\). In a separated convex bornological vector space \(\left( F,\mathcal{B}\right) \) over such a field \(K\), a pair \(\left( \mathcal{V},b\right) \), where \(\mathcal{V}: \bigcup _{k\in\mathbb{N}}\mathbb{N}^{k}\rightarrow\mathcal{P}\left( F\right) \) and \(b: \mathbb{N}^{\mathbb{N}}\rightarrow\left\{ \left| K^{\times}\right| \right\} ^{\mathbb{N}}\) (here, \(\left| K^{\times}\right| :=\left\{ \left| r\right| \mid r\in K\diagdown\left\{ 0\right\} \right\} \)), is called a bornological web if the following conditions hold:
(1)
the image of \(\mathcal{V}\) consists of disks,
(2)
\(\mathcal{V}\left( \varnothing\right) =F,\)
(3)
\(\mathcal{V}\left( n_{0},\dots,n_{k} \right) \) is absorbed by \(\bigcup_{n\in\mathbb{N}}\mathcal{V}\left( n_{0},\dots,n_{k},n\right) \) for each \(\left( n_{0},\dots,n_{k}\right) \in\) \(\bigcup_{k\in\mathbb{N}}\mathbb{N}^{k},\)
(4)
for every \(s:\mathbb{N\rightarrow N}\), the series \(\sum_{k\in\mathbb{N}}\lambda\left( s\right) _{k}x_{k}\) (with \(\lambda\left( s\right) _{k}\in K\)) converges bornologically in \(F,\) whenever \(x_{k}\in\mathcal{V}\left( s\left( 0\right) ,\dots,s\left( k\right) \right) \) and \(\left| \lambda\left( s\right) _{k}\right| =b\left( s\right) _{k}\).
Then \(\left( \mathcal{V},b\right) \) determines on \(F\) a convex linear bornology, \(\mathcal{B}_{\left( \mathcal{V},b\right) }\), and \(\left( F,\mathcal{B} _{\left( \mathcal{V},b\right) }\right) \) is called a webbed convex bornological space. The main result of the paper is the following.
Theorem 2.11. Let \(E\) and \(F\) be separated convex bornological vector spaces, where \(E\) is complete and \(F\) is endowed with a bornological web \(\left( \mathcal{V},b\right)\). Then every linear map \(f: E\rightarrow F\) with bornologically closed graph is bounded with respect to the given bornology on \(E\), and \(\mathcal{B}_{\left( \mathcal{V},b\right) }\) on \(F\).
As consequences of this theorem, the author obtains generalized versions both of the closed graph theorems of Popa and De Wilde cited above. For the last one, the field \(K\) is supposed to be spherically complete.

MSC:

46A17 Bornologies and related structures; Mackey convergence, etc.
46A30 Open mapping and closed graph theorems; completeness (including \(B\)-, \(B_r\)-completeness)
46J05 General theory of commutative topological algebras
46A08 Barrelled spaces, bornological spaces
46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
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