For two general states | Ψ Ψ ⟩ ⟩ ν ν = | Ψ Ψ 0 ⟩ ⟩ ν ν ⊕ ⊕ | Ψ Ψ 1 ⟩ ⟩ ν ν ⊕ ⊕ | Ψ Ψ 2 ⟩ ⟩ ν ν ⊕ ⊕ ⋯ ⋯ = a | 0 ⟩ ⟩ ⊕ ⊕ ∑ ∑ i a i | ψ ψ i ⟩ ⟩ ⊕ ⊕ ∑ ∑ i j a i j | ψ ψ i , ψ ψ j ⟩ ⟩ ν ν ⊕ ⊕ ⋯ ⋯ , and | Φ Φ ⟩ ⟩ ν ν = | Φ Φ 0 ⟩ ⟩ ν ν ⊕ ⊕ | Φ Φ 1 ⟩ ⟩ ν ν ⊕ ⊕ | Φ Φ 2 ⟩ ⟩ ν ν ⊕ ⊕ ⋯ ⋯ = b | 0 ⟩ ⟩ ⊕ ⊕ ∑ ∑ i b i | ϕ ϕ i ⟩ ⟩ ⊕ ⊕ ∑ ∑ i j b i j | ϕ ϕ i , ϕ ϕ j ⟩ ⟩ ν ν ⊕ ⊕ ⋯ ⋯ the inner product on F ν ν ( H ) is then defined as ⟨ ⟨ Ψ Ψ | Φ Φ ⟩ ⟩ ν ν := ∑ ∑ n ⟨ ⟨ Ψ Ψ n | Φ Φ n ⟩ ⟩ ν ν = a ∗ ∗ b + ∑ ∑ i j a i ∗ ∗ b j ⟨ ⟨ ψ ψ i | ϕ ϕ j ⟩ ⟩ + ∑ ∑ i j k l a i j ∗ ∗ b k l ⟨ ⟨ ψ ψ i | ϕ ϕ k ⟩ ⟩ ⟨ ⟨ ψ ψ j | ϕ ϕ l ⟩ ⟩ ν ν + ⋯ ⋯ where we use the inner products on each of the n -particle Hilbert spaces. Note that, in particular the n particle subspaces are orthogonal for different n .