= Anchored variation norm
{title2=$\|f\|_V=V_0^1(f),\quad f(0)=0$}
On the <vector space> of <continuous functions> $f:[0,1]\to\mathbb R$ of <bounded variation> with $f(0)=0$, the <total variation of a function> is a <norm>. Its <triangle inequality> and <absolute homogeneity of a norm> follow by applying the corresponding <absolute value> identities to every <partition of an interval>, then taking the <supremum>. The <partition of an interval> $(0,t,1)$ gives $|f(t)|\leq V_0^1(f)$, so zero <total variation of a function> forces $f=0$. The same inequality gives $\|f\|_\infty\leq\|f\|_V$, proving that convergence for this <norm> implies <uniform convergence>. Without $f(0)=0$, every constant <function> would have zero <total variation of a function>.
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