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?



Parseval frame, convergence of $\sum_{k=0}^\infty \left\|g_k\right\|$
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.


