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This implies dµλ2 Jλ = dµ where the right hand side is the generalized Radon Nikodym derivative (see Example 7 for this concept). The fact that Jλ ∈ / L2 (µ) for λ = 1 is in agreement with the fact that µλ2 and µ are singular measures if λ = 1. Example 13 (A simple second quantized operator) Let z ∈ C and Φ ∈ (N )′. Then SΦ has an entire analytic extension and we may consider the function θ → SΦ(zθ) , θ ∈ NC′ . 2 This function is also an element of Emax (NC′ ). Thus we may define Γz Φ by S (Γz Φ) (θ) = SΦ(zθ) .

KoSa78, Ko80a, Ko80b, KT80, HKPS93, BeKo88, KLPSW94]), while in this section we concentrate on the smallest space (N )1 . −1 2 1 Let (H−p )−1 be the dual with −q be the dual with respect to L (µ) of (Hp )q and let (N ) 1 2 respect to L (µ) of (N ) . We denote by . , . the corresponding bilinear dual pairing which is given by the extension of the scalar product on L2 (µ). We know from general duality theory that (N )−1 = ind lim (H−p )−1 . , for any Φ ∈ (N )−1 there exist p, q ∈ N such that Φ ∈ (H−p )−1 −q .

2 p,q , ✷ From general duality theory on nuclear spaces we know that the dual of (N ) is given by (N )′ = ind lim (H−p )−q , p,q∈N where (H−p )−q = (Hp )′q . We shall denote the bilinear dual pairing on (N )′ × (N ) by ·, · : Φ, ϕ = ∞ n! Φ(n) , ϕ(n) , n=0 where Φ ∈ (H−p )−q corresponds to the sequence (Φ(n) , n ∈ N) with Φ(0) ∈ C, and Φ(n) ∈ ⊗n HC,−p , n ∈ N. Remark. Consider the particular choice N = S(IR). , [HKPS93, PS91]. For the norms ϕ p ≡ Γ(Ap )ϕ 0 introduced there, we have · p = · p,0, and · p,q ≤ · p+ 2q .

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