Hopefully, I'm using the correct terms/names of things, mainly because the language in which I study is not English.
Given the operator $T$, in this case is the derivative operator , with the inner product: $$ <f,g>=\int_{a}^{b}f(x)\cdot g(x)dx $$ in vector space $V=\mathbb{R}_2[x]$ (polynomials with degree $\leq2$, in $[0,1]$)
Find the conjugate transpose operator $T^\ast$.
I'm aware that $T^\ast$ allows: $<T(v),u>=<v,T^\ast(u)>$ for all $u,v\in V$
After trying to play with the equation, I've got this:
let $f,g\in V$ such that: $$ <T(f),g>=. . .=f\cdot g-<f,T(g)>=<f,T^\ast(g)> $$
At this point I'm not sure how to extract $T^\ast$, so any direction of though is helpful.
Thanks!