Show that in an infinite dimensional Hilbert space $\displaystyle H$ the closed unit ball $\displaystyle (B(H))_{1}$ in not compact in the strong operator topology
Show that in an infinite dimensional Hilbert space $\displaystyle H$ the closed unit ball $\displaystyle (B(H))_{1}$ in not compact in the strong operator topology
Pick an infinite sequence of orthonormal elements. Work out the distance between them and so conclude that it cannot have a Cauchy subsequence. Every convergent sequence is Cauchy, so this means that it cannot have a convergent subsequence.