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What does it mean by "converges boundedly"?

This should mean that $U_{\mu_k} \to U$ almost everywhere on $\mathbb{R}^m \times [0,T)$, and moreover the sequence of functions $U_{\mu_k}$ is uniformly bounded: $$\sup_k \sup_{(x,t) \in \mathbb{R}^m …
Nate Eldredge's user avatar