Hölder's inequality for sequences.

[ELESSAR TELKONT]

Homework Statement

Let 1\leq p,q that satisfy p+q=pq and x\in\ell_{p},\, y\in\ell_{q}. Then
<br /> \begin{align}<br /> \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}}<br /> \end{align}<br />

Homework Equations

The Hölder's inequality for \mathbb{R}^{n} and convergence conditions of sequences in \ell_{r}, that is:
<br /> \begin{align}<br /> \sum_{k=1}^{\infty}\left\vert x_{k}\right\vert^{r}&lt;\infty<br /> \end{align}<br />

The Attempt at a Solution

I can prove the result from the inequality for \mathbb{R}^{n}, but I have a missing part that I don't get to prove, that is: proving that
<br /> \begin{align}<br /> \sum_{k=1}^{\infty}\left\vert x_{k}y_{k}\right\vert<br /> \end{align}<br />
converges given convergence conditions over x, y. Could you give me ideas! This is not a homework task. I'm reviewing some analysis topics.

 

[micromass]

How did you prove the inequality for \mathbb{R}^n?? Can you adapt the proof?

 

[ELESSAR TELKONT]

The problem I have is not the proof itself, but the convergence of the LHS of the inequality. How can I prove it is the question. Obviously in \mathbb{R}^{n} you don't need to check any convergence, then you have no manner to parallel that part of the proof.

In other words: I have followed the proof for \mathbb{R}^{n} and proven the inequality for sequences, but I failed to justify why I can do it since I don't know how to prove that if the series for x, y converge with the convergence condition for that sequence spaces then the series in LHS converges.

 

[ELESSAR TELKONT]

The result for \mathbb{R}^{n} is

Let 1\leq p,q that satisfy p+q=pq and x,y\in\mathbb{R}^{n}. Then
<br /> \begin{align}<br /> \sum_{k=1}^{n}\left\vert x_{k}y_{k}\right\vert\leq\left(\sum_{k=1}^{n}\left\vert x_{k}\right\vert^{p}\right)^{\frac{1}{p}}\left( \sum_{k=1}^{n}\left\vert y_{k}\right\vert^{q}\right)^{\frac{1}{q}}<br /> \end{align}<br />

 

[micromass]

Maybe you can start by proving that

\frac{|x_ny_n|}{\|(x_n)_n\|_p\|(y_n)_n\|_q}\leq \frac{1}{p}\frac{|x_n|^p}{\|(x_n)_n\|_p}+\frac{1}{q}\frac{|y_n|^q}{\|(y_n)_n\|_q}

In general if 0&lt;\lambda &lt;1 and a,b are nonnegative, then

a^\lambda b^{1-\lambda}\leq \lambda a+(1-\lambda)b

 

[ELESSAR TELKONT]

Why I can't see that! that's another version of the Young's inequality. thanks for that illuminating idea.