Possible Duplicate:
Compactness of Multiplication Operator on $L^2$
Let $u: \mathbb{R}\rightarrow \mathbb{C}$ be a bounded continuous function. Show that the multiplication operator $M_u$ defined on $L^2(\mathbb{R)}$ by $M_uf =uf$ cannot be compact on $L^2(\mathbb{R)}$ unless $u = 0$. And also that the same holds when $u\in L^\infty(\mathbb{R})$
I cannot think of some clever way of choosing the compact functions from the unit ball, such that there is no convergent subsequence of $(u(x)f_n(x))$