If we look at an $n$ - dimensional vector space $V$ and a linear transformation \begin{equation} T : V \to V, \quad x \mapsto Tx \quad \forall \, x \in V \end{equation} then given a choice of basis for $V$ one can represent $T$ in terms of a $n \times n$ matrix $A_T = A_T(i,j)$
Is this also the case for linear transformations on infinite - dimensional vector spaces, where we replace the matrix $A_T$ by an integral ?
In particular, since differentiation is a linear map, that would mean differentiation can be written in the form of an integral ... I realize this is either a dumb question (because it is obviously wrong) or it is some classic result I haven't found yet. In both cases it would be great to get some reference where I can learn more about it, many thanks!