Show that $\varphi$ is of class $C^{\infty}$ and compute $\varphi'$

Question

I am trying to solve the following question, arising from Implicit Function Theorem. $$\varphi :L(\Bbb{R}^n,\Bbb{R}^n)\to L(\Bbb{R}^n,\Bbb{R}^n)$$ $$A\mapsto\varphi(A)=A^2.$$

I want to show that $\varphi$ is of class $C^{\infty}$ and compute $\varphi'$

I need some help in doing these. I would have loved to define $C^\infty$ but based on the arguments about its definition in Why don't we define the class of $C^{\infty}$ in this way?, I think I don't even know which of them I should follow.

Anyway, I would greatly rely on your definition and guidance.

Answer 1

Each component of $\phi$ is a polynomial. Therefore $\phi\in C^{\infty}(L(\mathbb{R}^n),\mathbb{R}^n)$.

We need $\phi'(A_0)$ to satisfy $$\begin{align}0&=\lim_{A\to 0}\frac{\|\phi(A+A_0)-\phi(A_0)-\phi'(A_0)(A)\|}{\|A\|}\\&=\lim_{A\to 0}\frac{\|A^2+AA_0+A_0A-\phi'(A_0)(A)\|}{\|A\|}\\&=\lim_{A\to 0}\frac{\|AA_0+A_0A-\phi'(A_0)(A)\|}{\|A\|}\end{align}$$

Therefore, $\phi'(A_0)(A)=AA_0+A_0A$