Exercise 630Topology › Topology of Mn(ℝ)L1Added 4 Nov 2026

Convergence in $M_n(\mathbb{R})$ via entries

#matrix convergence#entrywise convergence#sequence of matrices

Statement

A sequence $(A^{(k)})_{k\ge1}$ of matrices in $M_n(\mathbb{R})$ is said to *converge entrywise* to $A\in M_n(\mathbb{R})$ if for every $i,j\in\{1,\ldots,n\}$ , the real sequence $(a^{(k)}_{ij})_{k\ge1}$ converges to $a_{ij}$ . Show that entrywise convergence is equivalent to convergence for the metric
$d(A,B)=\max_{i,j}|a_{ij}-b_{ij}|.$

Hint

Compare $d(A^{(k)},A)$ with the largest entrywise difference; use the fact that a maximum of finitely many quantities vanishes iff each one vanishes.

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Solution

$(\Leftarrow)$ . Suppose $d(A^{(k)},A)\to0$ . For each $(i,j)$ , $|a^{(k)}_{ij}-a_{ij}|\le d(A^{(k)},A)\to0$ , so entrywise convergence holds.

$(\Rightarrow)$ . Suppose entrywise convergence. For each $(i,j)$ , $|a^{(k)}_{ij}-a_{ij}|\to0$ . There are finitely many pairs $(i,j)$ , so
$d(A^{(k)},A)=\max_{i,j}|a^{(k)}_{ij}-a_{ij}|\to0.$

Conclusion. Entrywise convergence and convergence for the sup metric coincide on $M_n(\mathbb{R})$ .

Convergence in $M_n(\mathbb{R})$ via entries

Statement

Solution

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