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It just occurred to me that conjugate exponents, i.e. $p,q\in(1,+\infty)$ such that $$\frac{1}{p}+\frac{1}{q} =1$$ also satisfy the relations:

  • $\sum_{n=0}^\infty \frac{1}{p^n}=q;$
  • $\sum_{n=0}^\infty \frac{1}{q^n}=p.$

I never saw this fact used in the study of $L^p$ spaces... does anyone know any application of these relations in that context?

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1 Answer 1

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In the study of $L^p$ spaces, I think this implies, e.g., that a particular generalization of Hölder's inequality holds: Let $f_n \in L^{p^n}$ for $n \in \mathbb N$. Then

$$\left\Vert \prod_{n=0}^\infty f_n\right\Vert_{1/q} \leq \prod_{n=0}^\infty \Vert f_n \Vert_{p^n}.$$

This can be proven by passing to the limit here. I'm sure that one can construct more similarly artificial applications of this result.

Edit: It turns out that what I stated here is actually a special case of this answer.

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