Let $W^{1,1}_0(0,1)$ be the space of functions on the interval $(0,1)$ that vanish at the boundary with the standard $W^{1,1}(0,1)$-norm. $$||f|| = \int_0^1 |f(x)| \, dx + \int_0^1 |f'(x)| \, dx.$$

Is this space a Banach lattice? It appears to satisfy all the conditions listed here. Two functions $f, g$ satisfy $f < g$ if $f(x) < g(x)$ for all $x \in (0,1).$

What about the condition $$|| f || = || \, \, |f| \, \, ||$$? Is this obvious? Does one need to use smooth approximations?

3.If $f$ and $g$ are disjoint, then they satisfy $$ || f+g || = || f|| + ||g ||.$$ So is $W^{1,1}_0(0,1)$ an abstract $L^1$ space?

Any reference will be greatly appreciated.