# Closed unit ball in B(H)

#### Mauritzvdworm

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

#### Focus

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.