What exactly is the sobolev-space of $L^1$-valued functions?

Question

in a paper about PDEs, they use the Sobolev space:

$$W^{1,\infty}(0,T;L^1(0,L)),$$

where $T$ and $L$ are just positive real numbers. I know the concept of Sobolev spaces of functions which are real-valued (and also found something concerning vector-valued online), but I really have no idea what this space here is.

I would like to know:

1. What functions belong to this space?
2. How is the norm defined?
3. A source or a proof which states that this is a Banach space.

Thanks!

(The original paper can be found here. The above mentioned space appears in the beginning of the proof to theorem 3.1.)