About convergence in $L_\text{loc}^1$-norm

Question

I am confused by the expression below: $${||{u-u_m}||}_{L_\text{loc}^1(\Omega)}\to 0,\ \text{ as }\ m\to \infty.$$

Does it mean $$\lim_{m\to\infty}\sup_{\Omega'\subset\subset\Omega}{||{u-u_m}||}_{L^1(\Omega')}=0,$$ or $$\lim_{m\to \infty}{||{u-u_m}||}_{L^1(\Omega')}=0,\ \forall \Omega'\subset\subset \Omega?$$ Which one is correct? Or neither is correct?

Thank you for your help!

Answer 1

Convergence in $L^1_{\mathrm{loc}}$-topology is equivalent to the second statement (see here). I would never use the notation $\Vert \cdot \Vert_{L^1_{\mathrm{loc}}}$ because there is no such norm. (At most collection of seminorms turning it into a Frechét-space, respectively a metric that defines the topology - again see the wikipedia article).