= Completeness of the bounded-variation space
{title2=$BV(\Omega)\text{ is Banach}$}
A <Cauchy sequence> $(u_n)$ in the <BV space> converges in $L^1$ to $u$. For fixed sufficiently large $n$, <lower semicontinuity> gives $|D(u_n-u)|(\Omega)\le\liminf_m|D(u_n-u_m)|(\Omega)$. The $L^1$ <norm> of the same difference converges, so the small full-norm Cauchy bound passes to $u_n-u$. The <triangle inequality> then puts $u$ in the <BV space> and proves convergence in its full <norm>. This argument turns completeness of $L^1$ and <lower semicontinuity> of variation into completeness of the combined <norm>.
Back to article page
