$H$ and $K$ are Hilbert Spaces, $(u_n)$ and $(v_n)$ are sequences in $H$ and $K$ respectively. $\sum_{n=1}^{n=\infty} \|u_n\|\|v_n\| $ converges.
$T\colon H\rightarrow K$ is defined by $Tx=\sum_{n=1}^{\infty} \langle x,u_n\rangle v_n$.
I need to show that $T$ is compact, and I am frankly clueless. All I can think to say is that the first sum converging means each series is bounded, but I don't know if that is even relevant.
And hints/help would be appreciated.
Thanks