Proposition 3.9. in Serge Lang's Introduction to Differentiable Manifolds gives
Let $\mathbf{E}$, $\mathbf{F}$ be vector spaces which are isomorphic. If $u: \mathbf{E} \rightarrow \mathbf{F}$ is an isomorphism, we denote its inverse by $u^{-1}$. then the map $u\rightarrow u^{-1}$ from $Lis(\mathbf{E}, \mathbf{F}) \text{ to } Lis(\mathbf{F}, \mathbf{E})$ is a $C^{\infty}$-isomorphism. Its derivative at a point $u_0$ is the linear map of $L(\mathbf{E}, \mathbf{F}) \text{ to } L(\mathbf{F}, \mathbf{E})$ given by $$v\rightarrow u_0^{-1}vu_0^{-1}$$
$Lis$ and $L$ mean Linear isomorphism and Linear map, respectively.
I have the knowledge of implicit/inverse function theorem, which says $(f^{-1})'(y) = \frac{1}{f'\circ f^{-1}(y)}$. My main concern goes around the notations of $u, u_0 \text{ and } u_0^{-1}$. It would be nice of you to explain the proposition from the get-go.