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@YemonChoi In the reviesed one, I suppose M is any finite dimensional subspace of B instead of all of closed subspaces. The isomorphical isometry I wanna construct is between B/M and N in the question.
If $|h|^{1/p}(\int_\mathbb{R}\mathrm{M}(|gradw|^p)(x)dx)^{1/p}$ is $|h|^{1/p}(\int_\mathbb{R}(\mathrm{M}(|gradw|)(x))^pdx)^{1/p}$, everything is okay, But it is not...
