TITLE: Frobenius Norm Triangle Inequality
QUESTION [9 upvotes]: How can I go about proving the triangle inequality holds for the Frobenius norm?
I worked through $\|A+B\|_F \le \|A\|_F + \|B\|_F$ and was not able to make it work =/.

REPLY [13 votes]: You can use the fact that it's a norm  from inner product :
$$\langle A,B \rangle = \text{Trace}(A^TB)$$
Then:
$$\|A+B\|^2=\|A\|^2+\|B\|^2+2\langle A,B\rangle$$
using Cauchy-Schwarz inequalitie we have  : $$\langle A,B\rangle  \leq \|A\| \|B\|$$
and this gives:
$$\|A+B\|^2 \leq  \|A\|^2+\|B\|^2+2 \|A\| \|B\|= (\|A\| + \|B\|)^2$$