Talk:Convenient vector space

Move from Draft

No, this is not Original Research. This is a serious professionally written article with a good collection of references. But, maybe, too professional for a non-mathematical encyclopedia? Boris Tsirelson (talk) 20:11, 21 April 2015 (UTC)

"stops to be jointly continuous at the level of normed vector spaces" — is this a correct English? As far as I understand, for normed vector spaces it IS jointly continuous, but for more general spaces it is not. Boris Tsirelson (talk) 20:22, 21 April 2015 (UTC)

@Tsirel: please adopt this abandoned draft and take it through to become an article. Fiddle Faddle 20:43, 21 April 2015 (UTC)

I see here that we have better candidates to this job. Boris Tsirelson (talk) 21:00, 21 April 2015 (UTC)

Please feel free to encourage one. I have suppressed the automatic 6 months deletion of a stale draft for, well, 6 more months Fiddle Faddle 22:14, 21 April 2015 (UTC)

Rewrite of lead section of this article

The lead section in an article on a complex subject is supposed to be understandable by an educated person who is not expert in the field. I fall into this category, since I have an undergraduate math degree of some antiquity. I am suggesting here an alternative lead paragraph about this topic. If it's a mess, mathematicians can conclude that the original was not understandable, and fix it up. Here goes:

Title: Convenient vector spaces

In mathematics, Convenient vector spaces are locally convex vector spaces satisfying a very mild completeness condition. Mappings between them are smooth or C^∞
if they map smooth curves to smooth curves. This leads to a cartesian closed category of smooth mappings (see property 6 below).

A specialized type of calculus, aptly termed convenient calculus, is used in the analysis of convenient vector spaces. Although this type of calculus is not useful in solving equations, in many situations it is effective in helping mathematicians to recognize smooth mappings.

Traditional differential calculus is effective in the analysis of finite-dimensional vector spaces and for Banach spaces. Beyond Banach spaces, difficulties begin to arise; in particular, composition of continuous linear mappings stop being jointly continuous at the level of Banach spaces, for any compatible topology on their space.

An example, at the level of normed vector spaces, can be seen when evaluating ev: E × E^∗
→ ℝ where E, a locally convex vector space. E^∗
is its dual of continuous linear functionals equipped with any locally convex topology such that the evaluation mapping is separately continuous. If the evaluation is assumed to be jointly continuous, then there are neighborhoods U ⊆ E and V ⊆ E^∗
of zero such that ev(U × V) ⊆ [0,1]. However, this means that U is contained in the polar of the open set V; so it is bounded in E. Thus E admits a bounded neighborhood of zero, and is thus a normed vector space.

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Not sure where to reply. This looks good too me. I've made completeness condition a blue link. I suggest we go with this, and that any further edits can happen in article space (with the title "convenient vector space"). Sławomir Biały (talk) 11:19, 22 April 2015 (UTC)

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The rewritten introduction look good to me, but I would take the last paragraph and make it a footnote for "stops being jointly continuous at the level of normable spaces" [Footnote marker here]. I do not know how to make this. Peter Michor 12:19, 23 April 2015 (UTC) 11:33, 22 April 2015 (UTC)

Okay, I've moved the rewritten text to the article, and created a note as indicated. Thanks for fixing the link; I wasn't sure which of the many "completeness" pages was relevant. I've moved the draft to article space. —Anne Delong (talk) 13:20, 22 April 2015 (UTC)

Some context would be nice...

Now that the article is in mainspace, it could use some additions to make it more encyclopedic and less like a math lesson. For example, who first coined the term "convenient vector space", and where and when was it first used? Has the concept led to any major advancements? etc. Just something to think about... —Anne Delong (talk) 21:29, 22 April 2015 (UTC)

I have rewritten parts of the introduction. Please have a look. The term of convenient vector space was coined by Frolicher and Kriegl in the book (cited), following the lead of Steenrod (see property 6 in the text). Major advances are explained in the section Application: Manifolds of mappings between finite dimensional manifolds. Here proofs which before needed dozens of pages of estimates are reduced to a few lines, which are given in the text. I could give references there to illustrate this -- but I think it would be too much. I will add some references to other applications later. By the way, 'convenient topological vector spaces' redirected to 'locally convene vector spaces' I changed it to redirect to 'Convenient vector spaces'. Peter Michor 08:55, 23 April 2015 (UTC) — Preceding unsigned comment added by Pwm86 (talk • contribs)

Peter, please sign your messages (on talk pages) by four tildas: ~~~~. Boris Tsirelson (talk) 09:15, 23 April 2015 (UTC)