Polynomial maps between matrix spaces are continuous

Idea. All norms on $M_n(\mathbb{R})\cong\mathbb{R}^{n^2}$ are equivalent, so "continuous" means the usual continuity on $\mathbb{R}^{n^2}$. A polynomial in the coordinates is continuous, and a map into $M_m$ is continuous iff each of its $m^2$ entries is.

Step 1: coordinate functions. The map $\pi_{ij}:A\mapsto a_{ij}$ is linear, and $|a_{ij}|\le\|A\|_F$, so $|\pi_{ij}(A)-\pi_{ij}(B)|\le\|A-B\|_F$: it is $1$-Lipschitz, hence continuous.

Step 2: polynomials. Constants and the $\pi_{ij}$ are continuous. Sums and products of real continuous functions are continuous (limit laws). Every polynomial in the $a_{ij}$ is obtained from these by finitely many sums and products, so it is continuous.

Step 3: vector-valued maps. If $F:M_n\to M_m$ has entries $F_{kl}$ that are polynomials, then $\|F(A)-F(B)\|_F^2=\sum_{k,l}|F_{kl}(A)-F_{kl}(B)|^2$ tends to $0$ as $B\to A$, since each term does.

Step 4: the examples.
- $\operatorname{tr}A=\sum_i a_{ii}$ is linear.
- $A\mapsto A^T$ only permutes the entries.
- $A\mapsto A^k$: by induction, the entries of $A^k=A\cdot A^{k-1}$ are $\sum_l a_{il}(A^{k-1})_{lj}$, polynomials in the $a_{ij}$.
- $\det A=\sum_{\sigma\in S_n}\operatorname{sgn}(\sigma)\prod_i a_{i\sigma(i)}$ is a polynomial (Leibniz formula).
- $\operatorname{adj}A$ has entries $(-1)^{i+j}\det A^{(j,i)}$, where $A^{(j,i)}$ is $A$ with row $j$ and column $i$ removed: these are determinants of smaller matrices whose entries are entries of $A$, hence polynomials.

Remark. Polynomial maps are even *locally Lipschitz*, with $C^\infty$ regularity. The converse fails: $A\mapsto\|A\|$ and $A\mapsto\sqrt{\ }$ are continuous without being polynomial. The next problems treat the first genuinely non-polynomial map: the inverse.