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.