- #1
camillio
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Hello, I'm reading Gaussian measures on Hilbert spaces by S. Maniglia (available via google) and I have the following issue, regarding the proof.
He states in Lemma 1.1.4:
Let μ be a finite Borel measure on H. Then the following assertions are equivalent:
(1) \int_H |x|^2 \mu(dx) < \infty
(2) There exists a positive, symmetric, trace class operator Q s.t. for x,y \in H
<Qx, y> = \int_H <x,z><y,z> \mu(dz).
If (2) holds, then Tr Q = \int_H |x|^2 \mu(dx).
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The proof begins:
Spse (2) holds. Let (e_n)_{n\in \mathbb{N}} be an orthonormal basis of H. Then
<br /> \int_H |x|^2 \mu(dx) = \sum_{n=1}^\infty \int_H |<x, e_n>|^2 \mu(dx) = \sum_{n=1}^\infty <Qe_n, e_n> = Tr Q < \infty.<br />
What I have trouble with is the transitiono to the sum of <Qe_n, e_n>. If I suppose, that x, e_n may be complex, then I miss the adjoint part of the absolute value.
Most probably I miss some trivial notion, so any help will be appreciated.
He states in Lemma 1.1.4:
Let μ be a finite Borel measure on H. Then the following assertions are equivalent:
(1) \int_H |x|^2 \mu(dx) < \infty
(2) There exists a positive, symmetric, trace class operator Q s.t. for x,y \in H
<Qx, y> = \int_H <x,z><y,z> \mu(dz).
If (2) holds, then Tr Q = \int_H |x|^2 \mu(dx).
--------
The proof begins:
Spse (2) holds. Let (e_n)_{n\in \mathbb{N}} be an orthonormal basis of H. Then
<br /> \int_H |x|^2 \mu(dx) = \sum_{n=1}^\infty \int_H |<x, e_n>|^2 \mu(dx) = \sum_{n=1}^\infty <Qe_n, e_n> = Tr Q < \infty.<br />
What I have trouble with is the transitiono to the sum of <Qe_n, e_n>. If I suppose, that x, e_n may be complex, then I miss the adjoint part of the absolute value.
Most probably I miss some trivial notion, so any help will be appreciated.
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