# Raise an example of two matrix norms, their “mean” is not a matrix norm.

If $N(\bullet)$ and $M(\bullet)$ are two matrix norms, their mean is the function $M_n\to\mathbb{R}_{\ge0}$ $$P(\bullet)=\frac{N(\bullet)+M(\bullet)}{2}.$$

Here, a matrix norm is a vector norm on $M_n$ that satifies $N(xy)\le N(x)N(y)$.

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