The convergence of this infinite sum is important if F ν ν ( H ) is to be a Hilbert space. Technically we require F ν ν ( H ) to be the Hilbert space completion of the algebraic direct sum. It consists of all infinite tuples | Ψ Ψ ⟩ ⟩ ν ν = ( | Ψ Ψ 0 ⟩ ⟩ ν ν , | Ψ Ψ 1 ⟩ ⟩ ν ν , | Ψ Ψ 2 ⟩ ⟩ ν ν , … … ) such that the norm, defined by the inner product is finite ‖ ‖ | Ψ Ψ ⟩ ⟩ ν ν ‖ ‖ ν ν 2 = ∑ ∑ n = 0 ∞ ∞ ⟨ ⟨ Ψ Ψ n | Ψ Ψ n ⟩ ⟩ ν ν < ∞ ∞ where the n particle norm is defined by ⟨ ⟨ Ψ Ψ n | Ψ Ψ n ⟩ ⟩ ν ν = ∑ ∑ i 1 , … … i n , j 1 , … … j n a i 1 , … … , i n ∗ ∗ a j 1 , … … , j n ⟨ ⟨ ψ ψ i 1 | ψ ψ j 1 ⟩ ⟩ ⋯ ⋯ ⟨ ⟨ ψ ψ i n | ψ ψ j n ⟩ ⟩ i.e., the restriction of the norm on the tensor product H ⊗ ⊗ n