How is weak convergence in L^1(0,T; BV(\Omega)) defined?

Question

I have a question regarding a proof. Let $\Omega$ be an open bounded set of $\mathbb{R}^n$ and $T > 0$.

How is the following convergence defined? \begin{equation} U_n \longrightarrow u \text{ weakly in } L^1(0,T; BV(\Omega)), \end{equation} where $BV(\Omega)$ is the space of functions with bounded variation.

Answer 1

As always: It means $$ F(U_n) \to F(u) $$ for all $F$ in the (topological) dual space of $L^1(0,T;BV(\Omega))$.

However, the dual space of $L^1(0,T;BV(\Omega))$ seems to be quite ugly, therefore, this might not be what you intended.