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

Homework Equations

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

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

 

[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 $0<\lambda <1$ 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.