Matrix converge? says that "$A_n$ converges to $A$ means that for some matrix norm, $\left\lVert A_n−A\right\rVert \to 0$. Equivalently, it means that every entry of $A_n$ converges to the corresponding entry of $A$." Why are these equivalent (how can it be proved)?

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    $\begingroup$ All norms on a finite dimensional vector space are equivalent. math.mit.edu/~stevenj/18.335/norm-equivalence.pdf $\endgroup$ – Elliot G 2 days ago
  • $\begingroup$ So which norm would you use to show "each entry converges to the corresponding entry"? $\endgroup$ – Prasiortle 2 days ago
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    $\begingroup$ @Prasiortle $\|A\|=\max \{|a_{ij}|: 1\leq i,j \leq n\}$ would be one such norm. $\endgroup$ – Kavi Rama Murthy 2 days ago
  • $\begingroup$ Thanks very much! $\endgroup$ – Prasiortle 2 days ago

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