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I was reading the Wikipedia article on matrix norms, and it claims that for $A \in \mathbb{C}^{m \times n}$, we have $$\|A\|_2 \le \sqrt{\|A\|_1 \cdot \|A\|_{\infty}}$$ which is a "special case of Holder's Inequality."

I know a couple proofs of this matrix inequality, but I would like to solve it using Holder.

For vectors, Holder gives $$|x^*y| \le \|x\|_p\|x_q\| \hspace{1cm} p, q \ge 1 \text{ with } \frac 1p + \frac 1q = 1$$ and taking the limit as $p \to 1^+$ we have $$|x^*y| \le \|x\|_1 \|y\|_{\infty}.$$

But I haven't been able to apply a similar reasoning to the matrix norms.

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