(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let [itex]1\leq p,q[/itex] that satisfy [itex]p+q=pq[/itex] and [itex]x\in\ell_{p},\, y\in\ell_{q}[/itex]. Then

[itex]

\begin{align}

\sum_{k=1}^{\infty}\left\vert x_{k}y_{k}\right\vert\leq\left(\sum_{k=1}^{\infty}\left\vert x_{k}\right\vert^{p}\right)^{\frac{1}{p}}\left( \sum_{k=1}^{\infty}\left\vert y_{k}\right\vert^{q}\right)^{\frac{1}{q}}

\end{align}

[/itex]

2. Relevant equations

The Hölder's inequality for [itex]\mathbb{R}^{n}[/itex] and convergence conditions of sequences in [itex]\ell_{r}[/itex], that is:

[itex]

\begin{align}

\sum_{k=1}^{\infty}\left\vert x_{k}\right\vert^{r}<\infty

\end{align}

[/itex]

3. The attempt at a solution

I can prove the result from the inequality for [itex]\mathbb{R}^{n}[/itex], but I have a missing part that I don't get to prove, that is: proving that

[itex]

\begin{align}

\sum_{k=1}^{\infty}\left\vert x_{k}y_{k}\right\vert

\end{align}

[/itex]

converges given convergence conditions over x, y. Could you give me ideas! This is not a homework task. I'm reviewing some analysis topics.

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# Homework Help: Hölder's inequality for sequences.

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