I am reading Hui-Hsiung Kuo's "Gaussian measures in Banach spaces". In the page 59, there is the following exercise:
Let $D$ be a Hilbert-Schmidt operator and define $||x|| := |Dx|$, for all $x \in H$. Prove that $||\cdot ||$ is a measurable semi-norm.
(A semi-norm $||\cdot ||$ on $H$ is said to be measurable if $\forall \epsilon >0$ there is an orthogonal projection $P_0$ s.th. $dim P_0H < \infty$ and $\mu \{||Px||> \epsilon \}< \epsilon $ for all finite dimensional orthogonal projections $P\perp P_0$.)
To solve the problem I did the following:
I chose an arbitrary orthonormal basis $\{e_n\}$. We know that $\sum_{n=1}^{\infty} |De_n|^2 < + \infty$. For a fixed $\epsilon$, I chose the projection $P_0$ as the projection into the space $span\{e_1,\cdots,e_m\}$ for sufficiently large $n$ that we determine according to $\epsilon$. So we can make the sum $\sum_{n=m+1}^{\infty} |De_n|^2$ as small as possible. And for any $P\perp P_0$, $P_0 H \subset span\{e_{m+1},\cdots \}$.
But I couldn't do the rest. Does anyone have any suggestion?