Problem
Let $V$ be the vector space of real polynomials in one variable $X$ with degree at most $1$. Define the inner product $\langle f,g\rangle:=\int^{1}_{-1}f(t)g(t) \, dt$ on this space and the map $E:V\rightarrow V$ by $E(f)(X)=f(X+1)-f(X)$.
Find the adjoint map $E^{\star}$ of $E$.
Progress
EDIT:
We require $E^{\star}$ such that $\langle E(f),g\rangle=\langle f,E^{\star}(g)\rangle$.
We note that $\langle E(f),g\rangle =\int^1_{-1}[f(t+1)-f(t)]g(t)dt$.
$f,g \in V \implies f=a_f+b_ft$ and $g=a_g+b_gt$.
Thus, $\int^1_{-1}[f(t+1)-f(t)]g(t) \, dt=2b_fa_g$ I think.
So, $\langle f,E^{\star}(g)\rangle=2b_fa_g$, but how do we find $E^{\star}$ from this?
Any help would be very appreciated. Thanks.