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Let $1\leq p,q\le\infty$, let $\mathbb F=\mathbb R$ or $\mathbb C$, and let $d$ be a positive integer. Is it possible to show the equivalence of the norms $\left\|\cdot\right\|_p$ and $\left\|\cdot\right\|_q$ on $\mathbb F^d$ without resorting to Hölder's inequality?

Anne Bauval
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wjmolina
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  • It is actually the case that all norms are equivalent on finite dimensional linear spaces (not specifically $p$-norms). See: https://math.stackexchange.com/questions/57686/understanding-of-the-theorem-that-all-norms-are-equivalent-in-finite-dimensional – rubikscube09 Apr 29 '18 at 19:41

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Corollary 13.29. of this file shows this directly.

wjmolina
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    This kind of answer is not an answer https://meta.stackexchange.com/questions/225370/your-answer-is-in-another-castle-when-is-an-answer-not-an-answer – Anne Bauval Nov 13 '22 at 14:02