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?
Thanks in advance.
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?
Well, have you proved the finite case?
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.