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Define $ U := C^0 ([0,T], W^{1,2} ) \cap C^1 ([0,T] \cap L^2 ) \cap L^\infty ([0,T] , W^{s,2} ).$ Then how can I prove that $ \lim_{x_j \to \infty}| u |^2 = 0 $ by using the fact: $ C^1 ([0,T] , \mathcal S ) \text{ is dense in } U?$ Here $u : [0,T] \times \Bbb R^n \to \Bbb C^n$ , the norm for the space $U$ : $|u|_{s,T} := \sup _{t \in [0,T]} \| u(t) \|_{W^{s,2}}$, $W^{s,p}$ is the usual Sobolev space, $\mathcal S$ denotes the Schwarz class.

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    Do you see why you just have to show it when $u_j$ is in the Schwartz space?2012-10-29

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