I have a question which relate adjoint space. That is: ” Is $C^*[0,1]$ linear isomorphism with $C[0,1]$?” $C[0,1]$ is space of continous function in $[0,1]$. If we choose inner product of $C[0,1]$ is $<,>: C[0,1] \longrightarrow\Bbb{R}$, by $=\int_0^1 f(x)g(x)dx$ then $C^*[0,1]$ $=$ $C[0,1]$ (linear isomorphism)?