In the proof of the existence of weak solutions to the NSE (*[Navier-Stokes Equations][1]* by Constantin and Foias, Chapter 8), the following argument is made:

>Let $u_m$ converges weakly to $u$ in $L^2(0,T;V)$ where 
$$
V=\overline{\{f\in (C_0^\infty(\Omega))^n\mid \nabla\cdot f=0\}}^{H^1(\Omega)}
$$ By extracting a subsequence, we may assume that $u_m(t_0)$ converges to $u(t_0)$ weakly in $V$ for all $t_0\in[0,T]\setminus E$ for some $E$ of Lebesgue measure $0$. 

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I don't understand why this is true. The statement would not be true when $V=\mathbb{R}$ according to the comments to a previous [question][2]. Would anyone point me to the related theorems in known references (or sketch the proof)?


  [1]: http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=Foias&s5=Navier-Stokes%20Equations&s6=&s7=&s8=Books&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=2&mx-pid=972259
  [2]: https://mathoverflow.net/q/229898/14319