Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2005 / 2

For affine polynomials $f(t)=a+bt$ and $g(t)=c+dt$, direct integration gives $⟨f,g⟩=2ac+2bd/3$. This is bilinear and symmetric, and $⟨f,f⟩=2a^2+2b^2/3>0$ for every nonzero $f$. Thus it is an inner product.

The adjoint of differentiation on affine polynomials is determined by $⟨Df,g⟩=⟨f,D^{\ast }g⟩$. Since $Df=b$, the left side is $2bc$. Writing $D^{\ast }g=A+Bt$, the right side is $2aA+2bB/3$. Equality for every $a,b$ forces $A=0$, $B=3c$, henceThis is not simply negative differentiation: there are no endpoint conditions here that would remove the boundary term in an integration by parts argument.