# Math Help - Closed unit ball in B(H)

1. ## Closed unit ball in B(H)

Show that in an infinite dimensional Hilbert space $H$ the closed unit ball $(B(H))_{1}$ in not compact in the strong operator topology

2. Originally Posted by Mauritzvdworm
Show that in an infinite dimensional Hilbert space $H$ the closed unit ball $(B(H))_{1}$ 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.