= Minimum Dirichlet energy of a linear interpolant
{title2=$\int_a^b|u'|^2=\int_a^b|s'|^2+\int_a^b|u'-s'|^2$}
For nodes $a=x_0<\cdots<x_n=b$ and prescribed real node values, let $s$ be the <linear spline> joining them. Among all absolutely continuous <functions> $u$ with the same node values and square-integrable derivative, $s$ uniquely minimizes the <Dirichlet energy> $\int_a^b|u'|^2$.
On each node interval, $s'$ is constant and $u-s$ vanishes at both endpoints. The <fundamental theorem of calculus> therefore gives $\int s'(u'-s')=s'[(u-s)(x_{i+1})-(u-s)(x_i)]=0$. Expanding the square and summing proves the displayed energy decomposition. Its final term is nonnegative; equality forces $u'=s'$ almost everywhere and the endpoint values fix the additive constant, proving uniqueness. In particular, the minimum is $\sum_i (u(x_{i+1})-u(x_i))^2/(x_{i+1}-x_i)$. If all interior node values are zero and there is at least one interior node, only the two endpoint intervals contribute to this minimum.
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