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 Apr 25 awarded Popular Question Apr 12 comment Prove that $K$ is compact $l^\infty$ is a normed space (metric space is enough), so compactness of $K$ is equivalent to the statement that every sequence in $K$ has a subsequence which converges to an element of $K$. Apr 12 answered Prove that $K$ is compact Apr 10 comment Proving that $f = 0$ if $\int fg = 0$ for all $g \in S$ @MehdiJafarniaJahromi This is dominated convergence. For a fixed Schwartz $g$, $g$ is bounded, and we may assume the $a_n$ are uniformly bounded above by $1$, so the whole thing is dominated by $|f|$. Apr 10 answered Proving that $f = 0$ if $\int fg = 0$ for all $g \in S$ Apr 9 answered f bounded by an integrable function implies f is measurable Apr 4 awarded Notable Question Mar 30 comment Show that $g$ is increasing over $I$ I mention it because if your interval $I$ is of the form $[a,0]$ then it does not contain any nontrivial interval of the form $[0,b]$, so the first part doesn't apply. Mar 30 comment Show that $g$ is increasing over $I$ In the case that your interval contains an interval of the form $[a,0]$, consider instead $[0,|a|]$ to get $x\in(0,|a|]$ and $h>0$ with $g(x+h) < g(x)$. But then since $g$ is odd we now have $-g(-x-h) < -g(-x)$ so that $g(-x) < g(-x-h)$ and again we find that $g$ is not increasing on $[a,0]$. Mar 30 comment Show that $g$ is increasing over $I$ Any nontrivial interval containing $0$ contains a smaller interval of the form $[0,b]$ or $[a,0]$ which is nontrivial. If $f$ isn't increasing on $[0,b]$, it will not be increasing on any bigger interval either. Mar 30 answered Show that $g$ is increasing over $I$ Mar 30 awarded Nice Question Mar 29 answered What is the difference between two function spaces Mar 29 answered The squeeze theorem in the sequence $u_n=(\frac{2n}{3n+1})^n$, $n \in \Bbb{Z^+}$ Mar 29 reviewed Leave Open Is $d(x,y)=|x-y|^2$ a distance on $\mathbb{R}$? Mar 29 reviewed Close Can every function be represented as polynomial Mar 29 reviewed Close optimization problem :using probability dynamic programming Mar 29 reviewed Leave Open Numerical derivative of function wrt natural log of variable (non-analytic) Mar 29 reviewed Leave Open How to count in 4 dimensions an infinite set of 3rd degree polynomials (with integer coefficients a,b,c,d) using Cantors diagonilaziation? Mar 29 comment Absolute continuity on limit Edited so you can correct your vote, thank you.