Comments on: Soft analysis, hard analysis, and the finite convergence principle http://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-and-the-finite-convergence-principle/ Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence Tao Sat, 25 May 2013 11:46:45 +0000 hourly 1 http://wordpress.com/ By: Analogies between finitary and infinitary math | Abstract Art http://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-and-the-finite-convergence-principle/#comment-230426 Mon, 20 May 2013 15:04:02 +0000 http://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-and-the-finite-convergence-principle/#comment-230426 […] [1.3] Soft analysis, hard analysis, and the finite convergence principle […]

]]> By: Frank http://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-and-the-finite-convergence-principle/#comment-230111 Sat, 18 May 2013 05:28:15 +0000 http://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-and-the-finite-convergence-principle/#comment-230111 Dear professor:
Sorry for my ignorance. In my opinion, the proof of the part (the infinite convergence principle implying the finite convergence principle) is imperfect. The compactness or completeness of the real line might as well be avoided, since the theorem, i.e, the monotone sequence of rational numbers on [0,1] is Cauchy as well. The compactness of [0,1]^\omega depends on that, I think.

]]> By: andrescaicedo http://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-and-the-finite-convergence-principle/#comment-230025 Fri, 17 May 2013 19:50:42 +0000 http://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-and-the-finite-convergence-principle/#comment-230025 Jennifer Chayes’s link is outdated. The current link should be http://research.microsoft.com/en-us/um/people/jchayes/

[Corrected, thanks - T.]

]]> By: Bird’s-eye views of Structure and Randomness (Series) | Abstract Art http://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-and-the-finite-convergence-principle/#comment-228784 Sat, 11 May 2013 06:05:12 +0000 http://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-and-the-finite-convergence-principle/#comment-228784 […] Analogies between finitary and infinitary math (adapted from “Soft analysis, hard analysis and the finite convergence principle“) […]

]]> By: The spectral proof of the Szemeredi regularity lemma « What’s new http://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-and-the-finite-convergence-principle/#comment-199142 Tue, 04 Dec 2012 01:35:50 +0000 http://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-and-the-finite-convergence-principle/#comment-199142 [...] with for all . Applying (3) and the pigeonhole principle (or the finite convergence principle, see this blog post), we can find such [...]

]]> By: The three Moirai, continued « chorasimilarity http://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-and-the-finite-convergence-principle/#comment-195459 Fri, 23 Nov 2012 11:57:30 +0000 http://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-and-the-finite-convergence-principle/#comment-195459 [...] Now the construction is finished, let us let the Moirai to do their job. Finally, I shall recall my real goal, which I have never forgot. The real goal is to pass from understanding of the power of this lambda calculus sector of graphic lambda calculus to the real deal, called “computing with space”, namely to understand space from a computational perspective, not as a given receptacle, but as a small list of procedures along with some impossible to verify assertions (like that we may rescale indefinitely space), see “emergent algebras”, which can always be eliminated a posteriori, by a kind of finitization procedure. [...]

]]> By: Walsh’s ergodic theorem, metastability, and external Cauchy convergence « What’s new http://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-and-the-finite-convergence-principle/#comment-185892 Thu, 25 Oct 2012 18:10:45 +0000 http://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-and-the-finite-convergence-principle/#comment-185892 [...] which is useful in situations in which one does not expect a uniform convergence rate; see this previous blog post for some discussion of metastability. When interpreted in a finitary setting, this concept requires [...]

]]> By: Functional Analysis – Introduction. Part II « Noncommutative Analysis http://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-and-the-finite-convergence-principle/#comment-176965 Mon, 01 Oct 2012 09:02:42 +0000 http://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-and-the-finite-convergence-principle/#comment-176965 [...] of substance, which applies to concrete cases in analysis, that does not involve some kind of hard analysis, some epsilonaustics, some sweat. In our example, the hard analysis comes in the proof of the [...]

]]> By: Hard analysis, soft analysis — The Endeavour http://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-and-the-finite-convergence-principle/#comment-165073 Tue, 28 Aug 2012 15:18:47 +0000 http://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-and-the-finite-convergence-principle/#comment-165073 [...] I was trained as a bird, but I’ve become more of a frog over time. Of course these categories are not exclusive. Frogs sometimes act as birds and vice versa. The categories of hard and soft analysis are not exclusive either. See Terry Tao’s article on relating hard and soft analysis. [...]

]]> By: 254A, Notes 6: Ultraproducts as a bridge between hard analysis and soft analysis « What’s new http://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-and-the-finite-convergence-principle/#comment-93078 Sat, 15 Oct 2011 18:58:24 +0000 http://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-and-the-finite-convergence-principle/#comment-93078 [...] poor (particularly if the proof of (i) relied extensively on infinitary tools, such as limits). See this blog post for some related [...]

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