As I was reading the article Non linear elliptic and parabolic equations involving measure data, I came across with the following:
the sequence $\{f_n \}$ is bounded in the space $L^1(0,T;W^{-1,s})+L^1(0,T;L^1)$,
where $W^{-1,s}$ denotes the usual Sobolev space.
What does this mean exactly? I thought a possible interpretation is:
Consider $\{g_n\} \in L^1(0,T;L^1)$ then $f_n -g_n$ is bounded in $L^1(0,T;W^{-1,s})$
Is this true?
If $X$, $Y$ are Banach spaces, which continuously embed into some Hausdorff topological vector space $Z$, then $$ X+Y=\{h\in Z\mid \exists f\in X,\,g\in Y\colon h=f+g\} $$ and $$ \|h\|_{X+Y}=\inf\{\|f\|_X+\|g\|_Y\mid f\in X,g\in Y,h=f+g\}. $$ Boundedness in $X+Y$ then has the usual meaning.
The choice of $Z$ does not matter and is often swept under the rug. In your case you could take for $Z$ something like $L^1(0,T;L^1+W^{-1,s})$, which leaves with another sum of Banach spaces. For these you can use the embeddings $L^1, W^{-1,s}\hookrightarrow \mathcal{D}^\prime$.