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Finding the adjoint of an operator
Consider the vector space $C[0, 1]$ with inner product, \begin{align*} \langle f, g\rangle=\int_{0}^1f(t)g(t)\ dt. \end{align*} Let $T:C[0, 1]\rightarrow C[0, 1]$ the bounded linear operador given by,
\begin{align*} T(f)(t)=\int_{0}^tf(s)\ ds. \end{align*} How can I find the Hilbert adjoint operator $T^*$ of $T$?