Comments on: Question of the Day http://www.arsmathematica.net/archives/2008/01/04/question-of-the-day/ Dedicated to the mathematical arts. Tue, 02 Dec 2008 01:56:56 +0000 http://wordpress.org/?v=2.5.1 By: Todd Trimble http://www.arsmathematica.net/archives/2008/01/04/question-of-the-day/#comment-58165 Todd Trimble Tue, 08 Jan 2008 01:09:28 +0000 http://www.arsmathematica.net/archives/2008/01/04/question-of-the-day/#comment-58165 Walt: cool. (Uniform boundedness rules!) Walt: cool. (Uniform boundedness rules!)

]]> By: thw http://www.arsmathematica.net/archives/2008/01/04/question-of-the-day/#comment-58162 thw Mon, 07 Jan 2008 09:26:16 +0000 http://www.arsmathematica.net/archives/2008/01/04/question-of-the-day/#comment-58162 A : C([0,1]) -> C([0,1]) \oplus L¹([0,1]) is just the inclusion, while B: C([0,1]) -> C([0,1]) \oplus L¹([0,1]) is defined by B(f,g) := f + g. Then im A = ker pr_2 is closed and im B = L¹([0,1]) is also closed, but im BA = C([0,1]) \subset L¹([0,1]) is not closed (it is dense but it is not the whole set). A : C([0,1]) -> C([0,1]) \oplus L¹([0,1]) is just the inclusion, while B: C([0,1]) -> C([0,1]) \oplus L¹([0,1]) is defined by B(f,g) := f + g. Then im A = ker pr_2 is closed and im B = L¹([0,1]) is also closed, but im BA = C([0,1]) \subset L¹([0,1]) is not closed (it is dense but it is not the whole set).

]]> By: Walt http://www.arsmathematica.net/archives/2008/01/04/question-of-the-day/#comment-58160 Walt Mon, 07 Jan 2008 06:34:07 +0000 http://www.arsmathematica.net/archives/2008/01/04/question-of-the-day/#comment-58160 Todd, I now see why people work with the category you mention (Banach spaces with operators of norm at most 1). While thinking about this question, I've already reinvented that category three times. Todd, I now see why people work with the category you mention (Banach spaces with operators of norm at most 1). While thinking about this question, I’ve already reinvented that category three times.

]]> By: Jacob Freeze http://www.arsmathematica.net/archives/2008/01/04/question-of-the-day/#comment-58152 Jacob Freeze Sun, 06 Jan 2008 10:47:10 +0000 http://www.arsmathematica.net/archives/2008/01/04/question-of-the-day/#comment-58152 What this question needs is a little <em>mollification.</em> <a href="http://s262.photobucket.com/albums/ii97/JacobFreeze/?action=view&current=Heat_eqn.gif" rel="nofollow"></a> What this question needs is a little mollification.

]]> By: Rémy Oudompheng http://www.arsmathematica.net/archives/2008/01/04/question-of-the-day/#comment-58146 Rémy Oudompheng Sat, 05 Jan 2008 09:19:36 +0000 http://www.arsmathematica.net/archives/2008/01/04/question-of-the-day/#comment-58146 It would have closed image too, since it is an isomorphism onto its image, and the image being Banach makes it closed. It would have closed image too, since it is an isomorphism onto its image, and the image being Banach makes it closed.

]]> By: Walt http://www.arsmathematica.net/archives/2008/01/04/question-of-the-day/#comment-58144 Walt Sat, 05 Jan 2008 08:01:53 +0000 http://www.arsmathematica.net/archives/2008/01/04/question-of-the-day/#comment-58144 Oh, I see. I was imagining the first map was something more complicated, like f -> (f,f). Oh, I see. I was imagining the first map was something more complicated, like f -> (f,f).

]]> By: Doormat http://www.arsmathematica.net/archives/2008/01/04/question-of-the-day/#comment-58143 Doormat Sat, 05 Jan 2008 01:27:12 +0000 http://www.arsmathematica.net/archives/2008/01/04/question-of-the-day/#comment-58143 Walt: to answer your question to thw, the map E -> E \oplus F; x \mapsto (x,0) always has closed image. Erm, this seems kinda obvious to me??? Walt: to answer your question to thw, the map E -> E \oplus F; x \mapsto (x,0) always has closed image. Erm, this seems kinda obvious to me???

]]> By: Walt http://www.arsmathematica.net/archives/2008/01/04/question-of-the-day/#comment-58141 Walt Sat, 05 Jan 2008 00:01:45 +0000 http://www.arsmathematica.net/archives/2008/01/04/question-of-the-day/#comment-58141 thw: Why is the image of the first map closed? Todd: Your musing on my motivation is correct. thw: Why is the image of the first map closed?

Todd: Your musing on my motivation is correct.

]]> By: Walt http://www.arsmathematica.net/archives/2008/01/04/question-of-the-day/#comment-58140 Walt Sat, 05 Jan 2008 00:00:07 +0000 http://www.arsmathematica.net/archives/2008/01/04/question-of-the-day/#comment-58140 Robert Young: That is a reasonable request, except the question I'm asking is boringly technical. There's a laundry list of nice properties that maps between vector spaces have. Banach spaces are a kind of infinite dimensional vector space, but the natural notion of maps between them don't have nice properties. I was just musing on an idea to fix them up and make them nice again. I updated the post to link to Wikipedia's page on "abelian category", that that page itself is very technical. Robert Young: That is a reasonable request, except the question I’m asking is boringly technical. There’s a laundry list of nice properties that maps between vector spaces have. Banach spaces are a kind of infinite dimensional vector space, but the natural notion of maps between them don’t have nice properties. I was just musing on an idea to fix them up and make them nice again. I updated the post to link to Wikipedia’s page on “abelian category”, that that page itself is very technical.

]]> By: hellblazer http://www.arsmathematica.net/archives/2008/01/04/question-of-the-day/#comment-58138 hellblazer Fri, 04 Jan 2008 21:39:57 +0000 http://www.arsmathematica.net/archives/2008/01/04/question-of-the-day/#comment-58138 Further to Todd's first comment: Ban_1 may be the more pleasant category but arguably a lot of natural questions and phenomena belong more naturally in Ban. Neither category is `exact' in the sense of Barr, Bourn et al. and so looks very different from an abelian category, additivity or no additivity. This reminds me of something I keep meaning to ask the categorists: is there some kind of fibred category or enriched category way to `stratify' the morphisms of Ban in some sense according to their norm? Further to Todd’s first comment: Ban_1 may be the more pleasant category but arguably a lot of natural questions and phenomena belong more naturally in Ban. Neither category is `exact’ in the sense of Barr, Bourn et al. and so looks very different from an abelian category, additivity or no additivity.

This reminds me of something I keep meaning to ask the categorists: is there some kind of fibred category or enriched category way to `stratify’ the morphisms of Ban in some sense according to their norm?

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