In all formulation of Riesz-Thorin complex interpolation theorem I saw (e.g. here), it always involves function spaces like $L^p$. I would like to know if this theorem can be applied to linear operators on e.g. finite dimensional complex Hilbert space. I have seen this being used in several places, but they do not say why it works there.
As I am not very familiar with functional analysis, I would like to have some explicit arguments on how to pass from $L^p$ spaces, where the linear operators act on functions and then one evaluates the relevant $L^p$ norms, to complex Hilbert space $\mathbb{C}^d$ where the norms involved in the theorem for the complex interpolation should be replaced by Schatten $p$-norms for linear operators (which in this case would be some finite-dimensional matrices). My guess is that this should not be hard but I don't know how much functional analysis is needed to do this.