# Thread: How to prove Hölders inequality for sums?

1. ## How to prove Hölders inequality for sums?

Hi

I am trying to prove Hölders inequality for sums but I can't get any further. So far I have proven Young's inequality and then I got stuck. Can anyone help?

2. Originally Posted by Ase
Hi

I am trying to prove Hölders inequality for sums but I can't get any further. So far I have proven Young's inequality and then I got stuck. Can anyone help?

You need to be more specific. I assume you're talking about $\displaystyle \sum_{k=1}^{n}|x_ky_k|\leqslant\sqrt[\frac{1}{p}]{\sum_{k=1}^{n}|x_k|}\sqrt[\frac{1}{q}]{\sum_{k=1}^{n}|y_k|}$ with $\displaystyle \frac{1}{p}+\frac{1}{q}=1$. There is the alternative one for infinite series. Which are you after?

3. Take the measure theoric version of Hölder's inequality and apply it with a counting measure

4. Hi Drexel28

I'm sorry that I did'nt post a specifik formular (I am completely new to this forum and did'nt know how.) I am refering to the one for infinite series.

5. Originally Posted by Ase
Hi Drexel28

I'm sorry that I did'nt post a specifik formular (I am completely new to this forum and did'nt know how.) I am refering to the one for infinite series.

We know that for all fixed finite $\displaystyle m\in\mathbb{N}$ that $\displaystyle \sum_{n=1}^{m}|x_ny_n|\leqslant\left(\sum_{n=1}^{m }|x_n|\right)^{\frac{1}{p}}\left(\sum_{n=1}^{m}|y_ n|\right)^{\frac{1}{q}}$. Now apply the squeeze theorem and remember that we define $\displaystyle \sum_{n=1}^{\infty}a_n$ to be the limit of the partial sums.