$V=\mathbb{R}_{\leq2} [x]$ with the inner product: $\langle p(x),q(x) \rangle = p(0)q(0) + p(1)q(1) + p(-1)q(-1)$
Let $T: V \rightarrow V$ Linear operator as the following:
$T(cx^2 +bx +a) = cx^2 +(a-c+2b)x + (c-b)$
Find $T^*$.
So I wonder what is the most efficient way to that exercise? Should I find it according definition? or maybe I should find a orthonormal basis and find the matrix transformation of $T$ and from there I can get the matrix transformation of $T^*$.