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Clash Royale CLAN TAG #URR8PPP up vote 3 down vote favorite Is it generally true that $nablatimesvecn=0$ for any surface or is this only true for a simply connected domain? (see ftp://ftp.math.ucla.edu/pub/camreport/cam12-18.pdf) and discussion here (Curl of unit normal vector on a surface is zero?) I think Stoke’s theorem implies that $vecncdotnablatimesvecn=0$ but this isn’t quite $nablatimesvecn=0$. In particular, it doesn't seem like the unit vector for u in toroidal coordinates $(u,v,phi)$ satisfies this (http://mathworld.wolfram.com/ToroidalCoordinates.html) Yet constant u corresponds to toroidal surfaces. So what I'm pondering is how to translate $nablatimesvecn=0$ into practice. For example, the curl in general orthogonal curvilinear coordinates is beginalign nablatimesvecf=&frac1h_2,h_3,left[fracpartialpartial x_2left(h_3,f_3right)-fracpartialpartial x_3left(h_2,f_2right)right],vece_1+frac1...

Clash Royale CLAN TAG #URR8PPP up vote 1 down vote favorite Let $X,Y$ be Banach spaces and $T : X rightarrow Y$ a bounded operator that is weakly compact. How to prove that there exists $epsilon > 0$ and a sequence $(x_n)_n subset X$ such that $|x_n| = 1, |T(x_n)| geq epsilon$ forall $n in mathbb N$, and the sequence $(T(x_n))_n$ is congerging weakly to $0$ ? All what I can say is that I can extract a subsequence from $(Tx_n)_n$ which converges to a $y in Y$. Thanks for any help functional-analysis share | cite | improve this question asked Aug 20 at 13:01 Kébir J 25 5 I'm not sure you always can. In $Y=ell_1$, a sequence is weakly null if and only if it is norm null. – David Mitra Aug 20 at 13:09 Thank you David. This assersion was used in a paper wrote by William B. Johnson : eudml.org/doc/266092 (just under the first theorem in the first page). I ask my question here because I don...