Vector System State

Question

I have problems to bound a potential.

Let $w_i$ be vectors for $i = 1,\dots, n$ and $||w_i||$ the Euclidean norm of $w_i$.

$\Phi = 4 \sum_{i=1}^n ||w_i||^2$

$\Phi' = 4||w_1||^2 + 4||w_n||^2 + \sum_{i=2}^{n-1} ||w_{i-1} + w_{i+1}||^2 $

Does anyone of you see a chance to bound $\Phi'$ in terms of $\Phi$?

Thanks for your help :-)

Answer 1

We have that

$$\|x+y\|^2=<x+y,x+y>=\|x\|^2+\|y\|^2+2<x,y>\le 4\max\{\|x\|^2,\|y\|^2\}.$$ Using this we get

$$\|w_{i-1} + w_{i+1}\|^2\le 4 \max\{\|w_{i-1}\|^2,\|w_{i+1}\|^2\}\le \Phi.$$ Thus

$$\Phi' = 4||w_1||^2 + 4||w_n||^2 + \sum_{i=2}^{n-1} ||w_{i-1} + w_{i+1}||^2\le n\Phi.$$