Prove that $ \sum_{i=1}^{N} a_i \leq \sqrt{N \sum_{i=1}^{N}a_i^2} $ Prove that $ \sum_{i=1}^{N} a_i \leq \sqrt{N \sum_{i=1}^{N}a_i^2}  $. Well i choose $u=(1,\ldots,1)$ and $v=(a_1,\ldots,a_N)$ whit $a_i$ positive and. Apply $u \circ v \leq |u||v|$. With this now i want to prove that $\sum_{i,j}^{N}\frac{\partial u}{\partial x_i}\frac{\partial v}{\partial x_j}\leq N |\nabla u||\nabla v|$. I know that I am very close , but if somebody can help me, i will aprecciate..thank you

So we already know that $\sum_i a_i = u\circ v$. You then have $|u| = \sqrt{u\circ u} = \sqrt{N}$. And $|v| = \sqrt{v\circ v} = \sqrt{\sum_i a_i^2}$. Thus, $\sqrt{N\sum_i a_i^2} = |u||v|$, and so $u\circ v \leq |u||v|$ gives the desired result.
