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Let $\mu$ be a Borel probability measure on $[0, 1)$, and $\{g_k\}_{k=0}^\infty$ be a Parseval frame for $L^2(\mu)$. Does $$\sum_{k=0}^\infty \left\|g_k\right\|$$ converges?

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Any orthonormal basis of $L_2(\mu)$ is a Parseval frame which will cause the series to diverge. So, in general, the answer is no.

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