If the algebra of observables acts irreducibly on a Hilbert space, and if σ σ is a normal functional, that is, it is continuous in the ultraweak topology, then it can be written as σ σ ( ⋅ ⋅ ) = Tr ⁡ ⁡ ( ρ ρ ⋅ ⋅ ) with a positive trace-class operator ρ ρ of trace 1. This gives formula (5) above. In the case of a pure state, ρ ρ = | ψ ψ ⟩ ⟩ ⟨ ⟨ ψ ψ | is a projection onto a unit vector ψ ψ . Then σ σ = ⟨ ⟨ ψ ψ | ⋅ ⋅ ψ ψ ⟩ ⟩ , which gives formula (1) above.