I'm trying to find the adjoint operator of vector spaced defined by $V=\mathbb{P}_1[x]$, and $f:V\rightarrow V$
Now I have an inner product space defined by $\langle p,q \rangle$=$\frac{1}{2}\int_{-1}^{1}{p(t)}{q(t)}dt$ I also have that $f(p)=p(0)+p(1)t$ From this, I ought to be able to find and explicit form of $f^{*}$
So far, I have that $p(0)=a+b(0)$ and $p(1)=c+d(1)$ $\\$ So $\frac{1}{2}\int_{-1}^{1}{f(p)(x)}{q(t)}dt \rightarrow\frac{1}{2}\int_{-1}^{1}{(a+ct+dt)(x)}{q(t)}dt\rightarrow \frac{1}{2}\int_{-1}^{1}{(ax+ctx+dtx)}{q(t)}dt$
So I'm wondering if I'm missing an inner integral so that I could integrate to get an explicit form for $f^{*}$, thank you.