= Solution
For real $u\in L^1(\Omega)$, its <total variation seminorm on a domain> is
$$
\boxed{|Du|(\Omega)=\sup\left\{\int_\Omega u\,\operatorname{div}\xi\,dx:\xi\in C_c^1(\Omega;\mathbb R^2),\ |\xi(x)|_2\le1\right\}.}
$$
The <vector fields> have <compact support> inside $\Omega$, so no boundary term is charged. Taking both $\xi$ and $-\xi$ makes this equivalent to a supremum of absolute pairings. When finite, the <distributional derivative> $Du$ is a finite vector-valued <Radon measure>, and the displayed supremum is its <total variation norm of a measure>. For a <smooth> <function> it equals $\int_\Omega|\nabla u|_2\,dx$.
The <function of bounded variation on a domain> belongs to the <Banach space>
$$
\boxed{BV(\Omega)=\{u\in L^1(\Omega):|Du|(\Omega)<\infty\},\qquad\|u\|_{BV}=\|u\|_{L^1}+|Du|(\Omega).}
$$
The <L1 norm> is needed because variation alone vanishes on constant <functions> and is only a <seminorm>. No smoothness of $u$ is part of this definition.
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