The Frobenius norm comes from an inner product

Idea. The trace of $A^TB$ turns out to be the ordinary dot product of the two matrices read as vectors of length $n^2$. All the axioms of an inner product are then inherited from $\mathbb{R}^{n^2}$.

Step 1: the formula. The $(i,i)$ entry of $A^TB$ is $\sum_k (A^T)_{ik}B_{ki}=\sum_k a_{ki}b_{ki}$. Summing over $i$,
$$\operatorname{tr}(A^TB)=\sum_{i,k}a_{ki}b_{ki}=\sum_{i,j}a_{ij}b_{ij}.$$

Step 2: the axioms. Using this formula:
- *Bilinearity:* the right-hand side is linear in $A$ for fixed $B$, and linear in $B$ for fixed $A$.
- *Symmetry:* $\sum a_{ij}b_{ij}=\sum b_{ij}a_{ij}$.
- *Positivity:* $\langle A,A\rangle=\sum a_{ij}^2\ge0$, with equality iff every $a_{ij}=0$, i.e. $A=0$.

Step 3: the norm. $\sqrt{\langle A,A\rangle}=\big(\sum_{i,j}a_{ij}^2\big)^{1/2}=\|A\|_F$.

Consequence. The map $A\mapsto(a_{11},a_{12},\dots,a_{nn})$ is a linear bijection $M_n(\mathbb{R})\to\mathbb{R}^{n^2}$ which preserves inner products, hence distances. So $(M_n(\mathbb{R}),\|\cdot\|_F)$ is Euclidean space $\mathbb{R}^{n^2}$, and every notion of "open", "closed", "compact", "convergent" in $M_n(\mathbb{R})$ is the usual one of $\mathbb{R}^{n^2}$.

Remark. Because the Frobenius norm comes from an inner product, the Cauchy-Schwarz inequality $|\operatorname{tr}(A^TB)|\le\|A\|_F\|B\|_F$ and the parallelogram law hold in $M_n(\mathbb{R})$. We will use both repeatedly (for instance in the problems on submultiplicativity and strict convexity below).