Convergence in $L^{\infty}((0,T),L_{loc}^1(R))$ I studied a theorem, which says, that under certain conditions, using a numerical method, we can choose a subsequence, so that this subsequence converges to u(x,t) (the solution). The convergence is in the space $L^{\infty}((0,T),L_{loc}^1(R))$. I don't even know, what convergence in this space means.
Greetings
 A: In $L^1_{\rm{loc}}(\mathbb{R})$ space you have the following family of seminorms:
\begin{equation}
\lvert f\rvert_{[a, b]}=\int_a^b\, \lvert f(x)\rvert\, dx, 
\end{equation}
and convergence in $L^1_{\rm{loc}}(\mathbb{R})$ means convergence with respect to each one of them. The notation $L^\infty((0, T); L^1_{\rm{loc}}(\mathbb{R}))$ stands for the vector space 
\begin{equation}
L^\infty((0, T); L^1_{\rm{loc}}(\mathbb{R}))=\{F\colon (0, T)\to L^1_{\rm{loc}}(\mathbb{R}) \mid \forall a< b,\ \sup_{t \in (0,T)}\lvert F(t)\rvert_{[a, b]}< +\infty\},
\end{equation}
and convergence in this space means convergence with respect to each one of the seminorms 
\begin{equation}
\lvert F\rvert_{\infty, [a, b]}=\sup_{t \in (0, T)}\lvert F(t)\rvert_{[a, b]}, 
\end{equation}
that is, 
\begin{equation}
\begin{array}{lc}
F_n \to F\ \text{in}\ L^\infty((0, T); L^1_{\rm{loc}}(\mathbb{R})) \iff \\ \forall a <b,\ \int_a^b\lvert F_n(t, x)-F(t, x)\rvert\, dx\to 0\ \text{uniformly in}\ t \in (0, T).
\end{array}
\end{equation}
