The Quantization of Gravity by Claus Gerhardt

Author:Claus Gerhardt
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Proof

Since is nuclear, there exists a norm such that the embedding

(3.2.40)

is nuclear; i.e. we can write

(3.2.41)

where

(3.2.42)

(3.2.43)

and is an orthonormal sequence. We may, and shall, also assume

(3.2.44)

since D(A) is dense in H: Let

(3.2.45)

be a sequence of linearly independent vectors generating a dense subspace in H, then we can define an orthonormal basis in H which spans the same subspace. Hence, there exists a unitary map T such that

(3.2.46)

Instead of the embedding j, we can then consider the embedding

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