User banach - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T03:15:04Z http://mathoverflow.net/feeds/user/15334 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/120706/riesz-representation-for-an-infinite-dimensional-space Riesz representation for an infinite-dimensional space Banach 2013-02-03T21:10:18Z 2013-02-03T23:52:24Z <p>Suppose $X$ is an infinite-dimensional Banach algebra (hence not locally compact). </p> <p>Does there exist any sort of Riesz representation theorem that says something about elements of $C(X)^*$?</p> http://mathoverflow.net/questions/114059/absolutely-continuous-invariant-measures-for-piecewise-convex-maps/115337#115337 Answer by Banach for Absolutely Continuous Invariant Measures for Piecewise Convex Maps Banach 2012-12-03T21:22:43Z 2012-12-03T21:29:34Z <p>If $f$ is piecewise $C^2$, since your map is piecewise expanding it has an ACIM. See Lasota and York (Trans AMS, 1973). </p> <p>See also the <em><a href="http://books.google.it/books?id=2xsb6iveF9QC&amp;lpg=PA167&amp;ots=c7HDvka-L-&amp;dq=chaos%2520fractals%2520and%2520noise%2520asymptotically%2520stable%25206.2.2&amp;pg=PA148#v=onepage&amp;q=theorem%25206.2.2&amp;f=false" rel="nofollow">Chaos, Fractals, and Noise</a></em> by A. Lasota and M. Mackey. One of the theorems in Ch. 6 might work for your maps.</p> http://mathoverflow.net/questions/76474/applications-of-hilberts-metric/79207#79207 Answer by Banach for Applications of Hilbert's metric Banach 2011-10-26T23:00:44Z 2011-10-26T23:00:44Z <p>See e.g. pages 167-169 of "Topics in nonlinear analysis &amp; applications" By Donald H. Hyers, George Isac, Themistocles M. Rassias. You can read it on Google books.</p> http://mathoverflow.net/questions/4994/fundamental-examples/75461#75461 Answer by Banach for Fundamental Examples Banach 2011-09-14T23:46:00Z 2011-09-14T23:46:00Z <p>The Lorenz system of ordinary differential equations: $$\dot x=\sigma(y-x)$$ $$\dot y = rx-y-xz$$ $$\dot z = xy-bz$$ ($\sigma$, $r$, $b$ are parameters) is a good example in dynamical system. It is an example of a deterministic system displaying chaotic behaviour. Also the Lorenz attractor. Date: 1963.</p> http://mathoverflow.net/questions/69430/alternative-definitions-of-variation-of-a-function Alternative definitions of variation of a function Banach 2011-07-04T00:16:12Z 2011-07-04T10:39:13Z <p>Suppose $f \in L^1(\mathbb(R))$. Suppose $\int (\phi)(z):=\int_{x \leq z} \phi(x) dm(x)$.</p> <p>I am trying to understand if/why the following are equivalent:</p> <p>1) $\sup |\int f \phi \ dm|$, where sup is taken over all $\phi \in L^1$ with $\|\int(\phi)\|_\infty \leq 1$ and $\int \phi dm =0$. (G. Keller, Stochastic stability in some chaotic dynamical systems, Mh. Math 94, p.323)</p> <p>2) $\sup \int f g' dm$, where $g$ is $C^1(\mathbb{R})$ with compact support and $|g| \leq 1$.(E. Giusti, Minimal surfaces and function of bounded variation, Birkh\"auser)</p> http://mathoverflow.net/questions/120706/riesz-representation-for-an-infinite-dimensional-space/120719#120719 Comment by Banach Banach 2013-02-04T22:12:24Z 2013-02-04T22:12:24Z I looked up projective limit, but I don't see why $C(X)$ is the projective limit of $C(K)$. Would you please indicate what are the coordinate spaces and the bonding maps in the inverse limit? http://mathoverflow.net/questions/120706/riesz-representation-for-an-infinite-dimensional-space/120719#120719 Comment by Banach Banach 2013-02-04T20:43:15Z 2013-02-04T20:43:15Z According to my topology book, there is a restriction. The statement, &quot;a function on X is continuous iff its restrictions on compact subsets are continuous&quot; requires X to be compactly generated. However, in my case X is compactly generated. http://mathoverflow.net/questions/120706/riesz-representation-for-an-infinite-dimensional-space Comment by Banach Banach 2013-02-04T18:28:38Z 2013-02-04T18:28:38Z @Yemon, which smaller space do you suggest to consider instead of $C(X)$? Do you know of a reference for Riesz representation for $C(X)$ where $X$ is a metrizable non LCH space? http://mathoverflow.net/questions/120706/riesz-representation-for-an-infinite-dimensional-space/120719#120719 Comment by Banach Banach 2013-02-04T09:53:36Z 2013-02-04T09:53:36Z What does it mean that $C(X)$ is the projective limit of $C(K)$? Does this require X to be a countable union of compact sets? http://mathoverflow.net/questions/120706/riesz-representation-for-an-infinite-dimensional-space Comment by Banach Banach 2013-02-04T09:03:58Z 2013-02-04T09:03:58Z @Yemon, You are misunderstanding the question. I give you a Banach algebra $X$ that is infinite dimensional. I am asking for a topology on $C(X)$ and a characterization of its dual. I am asking if anyone has seen such sort of theorem. The focus is the infinite-dimensionality of $X$. Either you have seen a Riesz-type theorem for infinite-dimensional spaces or not. If you don't like $X$ being an algebra, assume it's just a Banach space, or assume it's just a topological vector space. http://mathoverflow.net/questions/120706/riesz-representation-for-an-infinite-dimensional-space Comment by Banach Banach 2013-02-03T22:50:04Z 2013-02-03T22:50:04Z $C(X)$ can be given a topology (most likely not a normable), but I don't want to specify one because that is part of the question. I am asking if there is some sort of Riesz representation theorem for an infinite-dimensional space $X$ where $C(X)$ has some nontrivial topology? http://mathoverflow.net/questions/120706/riesz-representation-for-an-infinite-dimensional-space Comment by Banach Banach 2013-02-03T22:10:52Z 2013-02-03T22:10:52Z Is $C(X)$ suppose to be a Banach space? http://mathoverflow.net/questions/120706/riesz-representation-for-an-infinite-dimensional-space Comment by Banach Banach 2013-02-03T21:27:56Z 2013-02-03T21:27:56Z I didn't say $C(X)$ is a Banach space. http://mathoverflow.net/questions/114059/absolutely-continuous-invariant-measures-for-piecewise-convex-maps/115337#115337 Comment by Banach Banach 2012-12-04T12:06:30Z 2012-12-04T12:06:30Z You can get away with $C^{1+\alpha}$, but with less regularity, I'm not sure. http://mathoverflow.net/questions/69430/alternative-definitions-of-variation-of-a-function Comment by Banach Banach 2011-07-04T14:21:15Z 2011-07-04T14:21:15Z Could someone please explain what is wrong with the following? Consider the function $f=0$ on $[0,1/2]$, $f=1$ on $[1/2,1]$. Obviously, $V(f)=1$. But applying the above definition we have: $$V(f)=\sup \{\int_{1/2}^1 g'(x) dx \}= \sup \{g(1)-g(1/2) \} \text{.}$$ Now take $g$ such that $g(1/2)=-1$ and $g(1)=1$ (here $\|g\|_\infty \leq 1$). It follows that $V(f) \geq 2$. http://mathoverflow.net/questions/69430/alternative-definitions-of-variation-of-a-function Comment by Banach Banach 2011-07-04T13:17:15Z 2011-07-04T13:17:15Z Thank you for the simple answer. For my future posts I will keep your second comment in mind. :)