We are given a power series with complex coefficients $\sum_{n \geqslant 0} a_n z^n$ with radius of convergence $R$ strictly positive, possibly equal to $+\infty$. We denote by $\|\cdot\|$ the norm associated with the inner product $(A,B)\mapsto\operatorname{Tr}({}^tA\times B)$ on $\mathcal{M}_d(\mathbb{R})$. Let $\mathcal{B} = \left\{A \in \mathcal{M}_d(\mathbb{R}), \|A\| < R\right\}$. Show that the map $\varphi : A \mapsto \varphi(A) = \sum_{n=0}^{+\infty} a_n A^n$ is defined and continuous on $\mathcal{B}$.
We are given a power series with complex coefficients $\sum_{n \geqslant 0} a_n z^n$ with radius of convergence $R$ strictly positive, possibly equal to $+\infty$. We denote by $\|\cdot\|$ the norm associated with the inner product $(A,B)\mapsto\operatorname{Tr}({}^tA\times B)$ on $\mathcal{M}_d(\mathbb{R})$.
Let $\mathcal{B} = \left\{A \in \mathcal{M}_d(\mathbb{R}), \|A\| < R\right\}$. Show that the map $\varphi : A \mapsto \varphi(A) = \sum_{n=0}^{+\infty} a_n A^n$ is defined and continuous on $\mathcal{B}$.