Let $E = L^2 (0, 1)$. Given $u ∈ E$, set $Tu(x)=\int_0^x u(t)dt$.
Solution says only $(u, T^* v) = \int_t^1 v(x)dx$.
I dont understand. adjoint operator is defined by $(Tu,v)=(u,T^*v)$. How do we get the formulae above?
Wew have $(Tu,v)=(u,T^*v)=\int_0^1 (\int_0^x u(t)dt)v(x)dx=\int_0^x u(t)(\int_0^1 v(x)dx)dt$ by interchanging the integrals but It looks like there is a variable change to do but I can't see where.
Thank you for any help/hints.