Limits of Matrix Operations, and the Closed Complete Sets of Symmetric and of Positive Semidefinite Matrices

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $m,n,p\in\mathbb{N}$, with the real matrices $\mathcal{M}_{r\times s}(\mathbb{R})$ for $r,s\in\mathbb{N}$ and $\mathcal{M}_{n}(\mathbb{R})=\mathcal{M}_{n\times n}(\mathbb{R})$, their entrywise operations, the matrix product, the matrix-vector product and the transpose $A^{\top}$; the distances $d_{r\times s}$ and $d_{n}$, metrics by The Real Matrices with the Operator-Norm Distance Form a Complete Metric Space with Entrywise Convergence §metric; the Euclidean distance $d_{E}$ on $\mathbb{R}^{q}$ for $q\in\mathbb{N}$, a metric by $\mathbb{R}^n$ with the Euclidean Distance Is a Complete Metric Space with Coordinatewise Convergence §metric, and the distance $d_{\mathbb{R}}$ on $\mathbb{R}$ of Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces §real-line; sequences and their convergence in a metric space; the set $\mathcal{S}(n)$ of symmetric matrices, and its subset $\mathcal{S}_{+}(n)$ of the positive semidefinite matrices in $\mathcal{S}(n)$ of The Invertible Matrices and the Positive Semidefinite Matrices Form Sets §psd; and the topology $\mathcal{T}_{d_{n}}$ of $(\mathcal{M}_{n}(\mathbb{R}),d_{n})$ as in Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces §topology, a set by The Open Sets of a Metric Space Form a Set, Given by a Predicative Formula §set.

¶ Let $(A^{(k)})$ and $(B^{(k)})$ be sequences in $\mathcal{M}_{m\times n}(\mathbb{R})$ converging to $A$ and $B$ in $(\mathcal{M}_{m\times n}(\mathbb{R}),d_{m\times n})$, $(C^{(k)})$ a sequence in $\mathcal{M}_{n\times p}(\mathbb{R})$ converging to $C$ in $(\mathcal{M}_{n\times p}(\mathbb{R}),d_{n\times p})$, $(t_{k})$ a sequence in $\mathbb{R}$ converging to $t$ in $(\mathbb{R},d_{\mathbb{R}})$, and $(x_{k})$ a sequence in $\mathbb{R}^{n}$ converging to $x$ in $(\mathbb{R}^{n},d_{E})$. Then the following hold.

¶ $(A^{(k)}+B^{(k)})$ converges to $A+B$ in $(\mathcal{M}_{m\times n}(\mathbb{R}),d_{m\times n})$.

¶ $(t_{k}A^{(k)})$ converges to $tA$ in $(\mathcal{M}_{m\times n}(\mathbb{R}),d_{m\times n})$.

¶ $(A^{(k)}C^{(k)})$ converges to $AC$ in $(\mathcal{M}_{m\times p}(\mathbb{R}),d_{m\times p})$.

¶ $(A^{(k)}x_{k})$ converges to $Ax$ in $(\mathbb{R}^{m},d_{E})$.

¶ $((A^{(k)})^{\top})$ converges to $A^{\top}$ in $(\mathcal{M}_{n\times m}(\mathbb{R}),d_{n\times m})$.

¶ $\mathcal{S}(n)$ is closed in the topological space $(\mathcal{M}_{n}(\mathbb{R}),\mathcal{T}_{d_{n}})$.

¶ $\mathcal{S}_{+}(n)$ is closed in $(\mathcal{M}_{n}(\mathbb{R}),\mathcal{T}_{d_{n}})$.

¶ Let $d_{\mathcal{S}(n)}$ be the restriction of $d_{n}$ to $\mathcal{S}(n)\times\mathcal{S}(n)$. Then $(\mathcal{S}(n),d_{\mathcal{S}(n)})$ is a complete metric space.

¶ Let $d_{\mathcal{S}_{+}(n)}$ be the restriction of $d_{n}$ to $\mathcal{S}_{+}(n)\times\mathcal{S}_{+}(n)$. Then $(\mathcal{S}_{+}(n),d_{\mathcal{S}_{+}(n)})$ is a complete metric space.