Exercise 1287Topology › Topology of Mn(ℝ)L1Added 4 Nov 2026

Completeness of $M_n(\mathbb{R})$

#completeness#Cauchy sequence#identification with R^{n^2}

Statement

Show that $M_n(\mathbb{R})$ equipped with the sup metric is complete.

Hint

Identify $M_n(\mathbb{R})$ with $\mathbb{R}^{n^2}$ ; use completeness of $\mathbb{R}$ .

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Solution

Identification. The map $\Phi:M_n(\mathbb{R})\to\mathbb{R}^{n^2}$ sending $A$ to its entries is an isometry for the sup metric.

Cauchy sequences. A Cauchy sequence in $M_n(\mathbb{R})$ corresponds to a Cauchy sequence in $\mathbb{R}^{n^2}$ . Since $\mathbb{R}^{n^2}$ is complete, it converges; identifying back gives convergence in $M_n(\mathbb{R})$ .

Conclusion. $M_n(\mathbb{R})$ is complete.

Completeness of $M_n(\mathbb{R})$

Statement

Solution

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