Talk:Bornology

Removed subsection; consistence and clarity issues

I removed from the article the subsection Characterizations, which ostensively claims to characterise locally bounded maps among linear maps between (topological) vector spaces. The claims would only hold with respect to sufficiently compatible bornologies on the vector spaces, but neither vector bornologies nor even the canonical / von Neumann bornology of a TVS have been discussed earlier in the article. The removed material (inculded below this comment) could be modified to fit a further expanded discussions of vector bornologies later in the article, but in its original location it was quite misleading and/or incomprehensible to people not knowledgeable of the topic.

In addition, the short section on bornologies on a topological vector spaces comes late and is (considering also the history of the subject) very short and limited. Further confusion is likely to arise from the terminology bounded sets appearing without qualification with two meanings, referring on the one hand to members of a given bornology on a set (or a vector space space) and on the other hand to the (von Neumann) bounded sets of topological vectors spaces, which form a particular (if very common) bornology on a TVS.

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Removed material:

Characterizations

Suppose that $X$ and $Y$ are topological vector spaces (TVSs) and $f:X\to Y$ is a linear map. Then the following statements are equivalent:

1. $f$ is a (locally) bounded map;
2. For every bornivorous (that is, bounded in the bornological sense) disk $D$ in $Y,$ $f^{-1}(D)$ is also bornivorous.

If $X$ and $Y$ are locally convex then this list may be extended to include:

3. $f$ takes bounded disks to bounded disks;

If $X$ is a seminormed space and $Y$ is locally convex then this list may be extended to include:

4. $f$ maps null sequences (that is, sequences converging to the origin $0$) into bounded subsets of $Y.$

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Stca74 (talk) 16:58, 1 April 2024 (UTC)