Derivative of polynomial from End(V) to End(V). In my differential geometry class I'm asked to prove that the map ${\rm End}(V) \to {\rm End}(V)$ that takes a function to a (fixed) polynomial in that function is differentiable. I think I have a good understanding of the definition of derivative and can often prove something is differentiable if I know what the derivative should be. But here I don't know how to proceed.
As ${\rm End}(\mathbb{R})$ is isomorphic to $\mathbb{R}$ I know the result for $V=\mathbb{R}$. But I can't see how to generalize this.
 A: Fix $p(t) \in \Bbb R[t]$ and define $p\colon {\rm End}(V) \to {\rm End}(V)$ by $T\mapsto p(T)$. This is one instance where it's easier to argue that $p$ is differentiable than describing the derivative ${\rm d}p_T\colon {\rm End}(V) \to {\rm End}(V)$.
The main thing to know here is that the composition $${\rm End}(V)\times {\rm End}(V) \ni (T,S) \stackrel{\circ}{\mapsto} T\circ S \in {\rm End}(V)\tag{1}$$is differentiable. Of course we're assuming that $\dim V < \infty$ here, so this differentiability follows from the matrix case: the entries of a matrix product $AB$ depend polynomially on the entries of each matrix $A$ and $B$.
For any $k\geq 1$, it follows that the map ${\rm End}(V) \ni T\mapsto T^k \in {\rm End}(V)$ is differentiable. Namely, for $k=1$ it is trivial, for $k=2$ it follows by composing $(1)$ with the diagonal embedding $${\rm End}(V) \ni T\stackrel{\Delta}{\mapsto} (T,T)\in {\rm End}(V)\times {\rm End}(V), \tag{2}$$which is differentiable. The induction step consists of noting that $T^{k+1} = T^k\circ T$ is a composition of differentiable functions, and hence is differentiable as well:
                                                       
Then we conclude it by noting that linear combinations of differentiable functions are differentiable.
As for what is the derivative? Consider $p(t) = a_0+a_1t+a_2t^2+a_3t^3$ as an example. Then $${\rm d}p_T(S) = a_1S + a_2(TS+ST) + a_3(T^2S + TST + ST^2),$$since compositions of endomorphisms is not a commutative operation.
