The Real Matrices with the Operator-Norm Distance Form a Complete Metric Space with Entrywise Convergence

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, for $r,s\in\mathbb{N}$ let $\mathcal{M}_{r\times s}(\mathbb{R})$ be the set of real matrices, with entries $A_{ij}$, the matrix-vector product $Ax$, the operator norm $\lVert A\rVert$ and the distance $d_{r\times s}$; let $\lVert x\rVert$ be the Euclidean norm on $\mathbb{R}^{q}$ for $q\in\mathbb{N}$, with the difference $x-y$ of $x,y\in\mathbb{R}^{q}$, and $d_{\mathbb{R}}$ the distance on $\mathbb{R}$ of Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces §real-line. For a sequence $(A^{(k)})$ in $\mathcal{M}_{r\times s}(\mathbb{R})$, $i\in[r]$ and $j\in[s]$, $(A^{(k)}_{ij})$ is the sequence $k\mapsto A^{(k)}_{ij}$ in $\mathbb{R}$ of The Sequences k ↦ f_k(p) of a Sequence in ℝ^I, and the Map of Their Limits §sections at the point $(i,j)$, as $\mathcal{M}_{r\times s}(\mathbb{R})=\mathbb{R}^{[r]\times[s]}$ by Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations §matrices. Convergence and Cauchy sequences are those of a metric space.

¶ For all $r,s\in\mathbb{N}$, $d_{r\times s}$ is a metric on $\mathcal{M}_{r\times s}(\mathbb{R})$, so that $(\mathcal{M}_{r\times s}(\mathbb{R}),d_{r\times s})$ is a metric space.

¶ For all $r,s\in\mathbb{N}$, $A,B\in\mathcal{M}_{r\times s}(\mathbb{R})$, $i\in[r]$ and $j\in[s]$, $|A_{ij}-B_{ij}|\le d_{r\times s}(A,B)$.

¶ For all $r,s\in\mathbb{N}$ and $A,B\in\mathcal{M}_{r\times s}(\mathbb{R})$, $\bigl|\lVert A\rVert-\lVert B\rVert\bigr|\le d_{r\times s}(A,B)$.

¶ For all $r,s\in\mathbb{N}$, $A,B\in\mathcal{M}_{r\times s}(\mathbb{R})$ and $x\in\mathbb{R}^{s}$, $\lVert Ax-Bx\rVert\le d_{r\times s}(A,B)\,\lVert x\rVert$.

¶ For all $r,s\in\mathbb{N}$, every sequence $(A^{(k)})$ in $\mathcal{M}_{r\times s}(\mathbb{R})$ and every $A\in\mathcal{M}_{r\times s}(\mathbb{R})$, $(A^{(k)})$ converges to $A$ in $(\mathcal{M}_{r\times s}(\mathbb{R}),d_{r\times s})$ if and only if $(A^{(k)}_{ij})$ converges to $A_{ij}$ in $(\mathbb{R},d_{\mathbb{R}})$ for all $i\in[r]$ and $j\in[s]$.

¶ For all $r,s\in\mathbb{N}$ and every sequence $(A^{(k)})$ in $\mathcal{M}_{r\times s}(\mathbb{R})$, $(A^{(k)})$ is a Cauchy sequence in $(\mathcal{M}_{r\times s}(\mathbb{R}),d_{r\times s})$ if and only if $(A^{(k)}_{ij})$ is a Cauchy sequence in $(\mathbb{R},d_{\mathbb{R}})$ for all $i\in[r]$ and $j\in[s]$.

¶ For all $r,s\in\mathbb{N}$, $(\mathcal{M}_{r\times s}(\mathbb{R}),d_{r\times s})$ is complete.