Hello, I'm reading Gaussian measures on Hilbert spaces by S. Maniglia (available via google) and I have the following issue, regarding the proof.(adsbygoogle = window.adsbygoogle || []).push({});

He states in Lemma 1.1.4:

Let μ be a finite Borel measure on H. Then the following assertions are equivalent:

(1) [itex] \int_H |x|^2 \mu(dx) < \infty[/itex]

(2) There exists a positive, symmetric, trace class operator Q s.t. for [itex]x,y \in H[/itex]

[tex] <Qx, y> = \int_H <x,z><y,z> \mu(dz).[/tex]

If (2) holds, then [itex]Tr Q = \int_H |x|^2 \mu(dx)[/itex].

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The proof begins:

Spse (2) holds. Let [itex](e_n)_{n\in \mathbb{N}}[/itex] be an orthonormal basis of H. Then

[tex]

\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.

[/tex]

What I have trouble with is the transitiono to the sum of [itex]<Qe_n, e_n>[/itex]. If I suppose, that [itex]x, e_n[/itex] 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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# Homework Help: Norm in Hilbert space

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