Consider the operator $T=i(x\partial_y -y\partial_x)$ with $i^2=-1$, [ or in polar coordinates $T=i\partial_{\theta},\quad (x,y)\to (r,\theta)$].
My question is, I want to construct a sequence $(u_n)$
$u_n\to 0$ weakly.
$\forall n\in \Bbb{N}:||u_n||=1$.
$||T u_n||\to 0$
were $||f||^2=\int_{\Bbb{R}^2} f(x,y)\overline{f(x,y)} dxdy $.
Note that for a radial function $f$ we have $T(f)=0$.