In the proof of the existence of weak solutions to the NSE (Navier-Stokes Equations 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 $0$.

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. Would anyone point me to the related theorems in known references (or sketch the proof)?

In the proof of the existence of weak solutions to the NSE (Navier-Stokes Equations 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 $0$.

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. Would anyone point me to the related theorems in known references (or sketch the proof)?