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Hölder's inequality for matrices
I was wondering if the Hölder's inequality was true for matrix induced norms, i.e. if
$$\|AB\|_1 \leq \|A\|_p\|B\|_q, \quad\forall p,q \in [1,\infty] \text{ s.t. } \tfrac{1}{p}+\tfrac{1}{q} = 1.$$
But …