|WikiProject Mathematics||(Rated Start-class, Mid-importance)|
I moved some talk that was here to Talk:Boundedness, since it was really about how to work (what became) that disambiguation page. This move frees this page for discussion specific to Bounded set. -- Toby Bartels 07:56, 2005 Mar 7 (UTC)
I moved some talk that was here to Talk:Bounded set (topological vector space), since it was really about that. This move frees this page for discussion specific to Bounded set in the basic meaning considered here. — MFH:Talk 21:35, 12 October 2006 (UTC)
Set is bounded in locally convex space iff bounded under each semi norm
Bounded linear operator is continuous ?
Hi Oleg, you reverted my deletion of this statement. As far as I know every continuous linear operator between locally convex spaces is bounded but the converse is not true in arbitrary locally convex spaces. It is true in semi-normed vector spaces. MathMartin 18:04, 19 Apr 2005 (UTC)
- You are very probably right.
- But now, you said Bounded set that
- The topology of a locally convex topological vector space can be defined by a family of semi norms.
- And here you say that every bounded linear operator is continuous in a seminormed space. Do the two imply that any linear bounded operator between locally convex spaces is bounded? Oleg Alexandrov 18:23, 19 Apr 2005 (UTC)
- The semi-norms in the familiy of semi-norms are not identical, that is just because the topology is defined using a familiy of semi-norms does not imply the space is semi-normable. In a semi-normable space the topology of the space can be described by 1 semi-norm (which is of course trivially a familiy of semi-norms). There are spaces ,called locally convex bornological spaces I think, in which boundedness is equivalent to continuity. Those spaces are nicer than locally convex spaces but not as nice as semi-normed spaces. MathMartin 20:03, 19 Apr 2005 (UTC)
- Got it, thanks. Oleg Alexandrov 20:25, 19 Apr 2005 (UTC)
Is this article too complicated ?
I don't plan to edit this article, partially because I don't know a lot of things about functional analysis. But, as a side remark, I think this article is too complicated.
Just from the very introductory paragraph one starts talking about metric spaces and topological vector spaces and the fine distinctions between them. Then, the very first section is called "Metric spaces". I think this should be inpenetrable to most undergraduates.
I sort of like the version of this article which was a while ago. First one started with a blurb about size, that was the introduction (no more). Then, the very first section was called "Bounded sets in calculus" or so, where one talked about the real line, sets bounded from below, above and both.
Then, I would put "Metric spaces" as its own section, just below. (So, the two headings "Simple definition" and "Genereal definition" would become standalone sections, one called "Calculus..." and one called "Metric spaces". I know this induces some repetition, but being most general and most concise is not always the best.
After that, the text could become as mathematical as one wishes.
The blurb about the fine differences between metric spaces and linear topological spaces could find itself a place in the discussion about linear topological spaces.
These are just some thoughts. Oleg Alexandrov 15:04, 2 May 2005 (UTC)
- The problems you mention are largely my fault I guess. I intend to split the article into bounded set and bounded set (topological vector space) (or something like this). After the somewhat complicated material about topological vector spaces has been removed the older structure could be restored.
- The reason I removed the previous Calculus heading was because a bounded set in calculus is only a special case of the more general definition in metric spaces, whereas the definition in topological vector spaces is completely different.MathMartin 15:53, 2 May 2005 (UTC)
- I understand your reasoning for removing the "Calculus" heading, I did not realize that in metric spaces and in linear topological spaces things are so different (but now I recall that, from a course I took a long time ago).
The opening line of the article is:
"In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size."
In no way, shape, or form does a bounded set imply a set be of finite size. It really gives the wrong impression. This should definitely be reworded. --anon
- And that's meant to be "informal description". Again, comments, suggestions, welcome. :) Oleg Alexandrov (talk) 00:09, 21 February 2006 (UTC)
Now that Bounded set (topological vector spaces) has its own article, it looks like this could naturally be split into Bounded set (metric spaces) and Bounded set (order theory). Boundedness of a set of real numbers is the only link between these two; maybe it should also have its own (expanded) article due its wide familiarity from calculus courses.
This is not really a proposal yet; I'd just like to see reactions.
Boundedness vs open intervals
So, an open interval of real numbers is bounded, am I correct? When searching maxima and minima over a convex set however it matters whether the edges are included or not. What is the n dimensional generalisation of the open/closed interval distinction called? --188.8.131.52 (talk) 13:58, 7 August 2009 (UTC)