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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