Adjoint of an operator on an inner product space Find the adjoint of 
$$\left ( Lf \right )\left ( t \right )=t^{2}\frac{df}{dt}+tf$$ where $$f\left ( 0 \right )=1$$ and$$ f'\left ( 1 \right )=2$$
under $$\left \langle f,g \right \rangle=\int_{0}^{1} f\left (t  \right )g\left ( t \right )dt$$
Working out I get 
$$\int_{0}^{1}\left ( t^{2}f'\left ( t \right )g+tf\left ( t \right )g\left ( t \right ) \right )dt$$
The 'second' integral that is the terms $$tf\left ( t \right )g\left ( t \right )$$ is a concern. Can I have some help?

The first term is given by integration by parts, the second one by noticing that
$$(tf(t))g(t) = f(t)(tg(t)).$$
Therefore,
$$\langle tf(t),g(t)\rangle = \langle f(t),tg(t)\rangle$$
so that the operator $f(t)\mapsto tf(t)$.
