= Adjoint of differentiation on affine polynomials
{title2=$D^*(c+dt)=3ct$}
Give the real <vector space> of degree-at-most-one <polynomials> the <inner product> $\langle f,g\rangle=\int_{-1}^1fg$. In coordinates $f=a+bt$, $g=c+dt$, this is $2ac+2bd/3$. Since $Df=b$, the adjoint identity requires $2bc=\langle a+bt,D^*(c+dt)\rangle$ for every $a,b$. Comparing the two coefficients yields the displayed formula. The <adjoint operator> depends on both the inner product and the domain; a formal integration-by-parts sign without the boundary term would give the wrong answer.
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