User josh - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T11:23:28Z http://mathoverflow.net/feeds/user/23320 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/95517/density-of-0-homogeneous-functions-in-h1-partial-omega Density of 0-homogeneous functions in $H^1(\partial \Omega)$ Josh 2012-04-29T19:37:40Z 2012-05-01T21:13:47Z <p><strong>Recall:</strong> A function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is called $0$-homogeneous if $f(\lambda x)= f(x)$ for every $\lambda>0$ and every $x\in \mathbb{R}^n$.</p> <p><strong>Question:</strong> Let $B$ a convex balanced and absorbent bounded domain of $\mathbb{R}^n$. Is the space of $0$-homogeneous $C^\infty(\mathbb{R}^n\setminus{0})$ functions dense in $H^1(\partial B)$?</p> http://mathoverflow.net/questions/95538/stable-subsets-with-respect-to-pointwise-convergence Stable subsets with respect to pointwise convergence. Josh 2012-04-30T00:28:24Z 2012-04-30T00:33:48Z <p>Consider the linear spacet $\mathcal{F}(\mathbb{R}^n)$ of all real functions defined in $\mathbb{R}^n$. It is well known that the subspace $\mathcal{C}(\mathbb{R}^n)$ of all real valued continuous function defined in $\mathbb{R}^n$ is stable with respect to the uniform (convergence) limit of elements in $\mathcal{C}(\mathbb{R}^n)$.</p> <p><strong>Question 1:</strong> Which is the <strong>smallest set</strong> (with respect to inclusion relation) containing $\mathcal{C}(\mathbb{R}^n)$ and stable with respect to <strong>pointwise convergence</strong>? </p> <p><strong>Question 2:</strong> Which is the <strong>smallest linear subspace</strong> of $\mathcal{F}(\mathbb{R}^n)$ which is stable with respect to <strong>pointwise convergence</strong>?</p> http://mathoverflow.net/questions/95517/density-of-0-homogeneous-functions-in-h1-partial-omega Comment by Josh Josh 2012-05-01T21:12:02Z 2012-05-01T21:12:02Z It depends on the know how. My research is in Abstract Algebra, and this question was just a curiosity (and I've never spent time in trying to prove what I'm asking). But you are right, cause I have spent so much time in replying to your uneuseful comments that It will be better for me to try to think about a proof whenever I will need this result. http://mathoverflow.net/questions/95538/stable-subsets-with-respect-to-pointwise-convergence Comment by Josh Josh 2012-04-30T12:21:22Z 2012-04-30T12:21:22Z Thankyou very much. My question started from a guess between the equivalence of &quot;what I now know to be the class of Baire functions&quot; and Borel functions. Dut to your answers I was able to find this www.jstor.org/stable/1996801. So thanks again.