Remark 2

The set $\mathscr{L}(\mathbf{R}^m; \mathbf{R}^n)$ of linear (necessarily continuous) mappings of $\mathbf{R}^m$ into $\mathbf{R}^n$ can be identified with the set $M_{n,m}(\mathbf{R})$ of matrices with $n$ rows and $m$ columns with coefficients in $\mathbf{R}$ and hence can be identified with $\mathbf{R}^{mn}$; on $\mathscr{L}(\mathbf{R}^m; \mathbf{R}^n)$, the uniformity of bounded convergence (with respect to the Euclidean metric on $\mathbf{R}^m$), of compact convergence, and of pointwise convergence are then identified with the additive uniformity on $\mathbf{R}^{mn}$. Take the norm of $x = (x_i) \in \mathbf{R}^n$ to be

and let $(e_j)$ be the canonical basis of $\mathbf{R}^m$; if $u$ and $v$ are two linear mappings of $\mathbf{R}^m$ into $\mathbf{R}^n$ such that $\|u(e_j) - v(e_j)\| \leq \epsilon$ for $1 \leq j \leq m$, then we have $|\alpha_{ij} - \beta_{ij}| \leq \epsilon$ for each pair $(i, j)$ [$U = (\alpha_{ij})$ and $V = (\beta_{ij})$ being the matrices of $u, v$ respectively]; and conversely, if these inequalities are satisfied, we have $\|u(x) - v(x)\| \leq ma\epsilon$ for every point $x$ of a cube of centre o and side $a$ in $\mathbf{R}^m$.