Exercise 1289Topology › Topology of Mn(ℝ)L1Added 4 Nov 2026

Equivalent metrics on $M_n(\mathbb{R})$

#equivalent metrics#norms on matrices#finite dimension

Statement

Consider the following three norms on $M_n(\mathbb{R})$ :
$\|A\|_\infty=\max_{i,j}|a_{ij}|,\qquad \|A\|_1=\sum_{i,j}|a_{ij}|,\qquad \|A\|_F=\left(\sum_{i,j}|a_{ij}|^2\right)^{1/2}.$
Show that they are equivalent.

Hint

Bound each by the others using the number of entries $n^2$ .

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Solution

Bounds. For every $A\in M_n(\mathbb{R})$ :
$\|A\|_\infty\le\|A\|_F\le\|A\|_1\le n^2\|A\|_\infty.$
- $\|A\|_\infty\le\|A\|_F$ : each entry squared is $\le\sum|a_{ij}|^2$ .
- $\|A\|_F\le\|A\|_1$ : from $\|A\|_F^2=\sum|a_{ij}|^2\le(\sum|a_{ij}|)^2$ .
- $\|A\|_1\le n^2\|A\|_\infty$ : sum of $n^2$ terms each $\le\|A\|_\infty$ .

Conclusion. The three norms define the same topology on $M_n(\mathbb{R})$ .

Equivalent metrics on $M_n(\mathbb{R})$

Statement

Solution

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