I'm having bit trouble in understanding weak convergences in Bochner space. I have following question for some general measurable space $\Omega$:

Let $\{x_n\}$ be a bounded sequence in $L^2((0,T)\times \Omega ; H^1(\mathbb{R}^d))$, then we can extract a weakly convergent sub sequence $\{x_{n_k}\}$ st $x_{n_k} \rightharpoonup x $ in $L^2((0,T)\times \Omega ; H^1(\mathbb{R}^d))$. Can we conclude for a.e. $t$, $x_{n_k}(t) \rightharpoonup x(t) $ in $L^2(\Omega ; H^1(\mathbb{R}^d))$?


(a) is not even true in ordinary Lebesgue spaces so it's certainly not true in Bochner. The standard counterexample is something like $x_n(t) = e^{2\pi i n t}$ in $L^2([0,1])$. It converges weakly to 0 but $x_n(t)$ diverges for almost every $t$.

In your specific context, you could take instead $x_n(t) = e^{2 \pi i n t} \phi$ where $\phi $ is some fixed nonzero element of $L^2(\Omega; H^1(\mathbb{R}^d))$.

  • $\begingroup$ Would this example also work for part b) ? $\endgroup$ – MathAnimal Sep 24 '18 at 4:54
  • $\begingroup$ Hello sir, could you please elaborate on this. $\endgroup$ – MathAnimal Sep 25 '18 at 14:04
  • $\begingroup$ @MathAnimal: Sorry, my idea was wrong. Maybe someone else will answer. $\endgroup$ – Nate Eldredge Sep 25 '18 at 14:33
  • $\begingroup$ Hi, shall I post this as a different question, this post seems to have saturated. $\endgroup$ – MathAnimal Sep 25 '18 at 15:43

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