there is $\displaystyle \{y_{1}..y_{n}\}$ a group of vectors in $\displaystyle R^{n}$ ,there is $\displaystyle \{u_{1}..u_{n}\}$ orthonormal basis of $\displaystyle R^{n}$.

the first part i have proved that for$\displaystyle \lambda_{1}..\lambda_{n}\$ in R

$\displaystyle \parallel\sum_{i=1}^{n}\lambda_{i}y_{i}\parallel^{ 2}\leq(\sum_{i=1}^{n}|\lambda_{i}|^{2})(\sum_{i=1} ^{n}\parallel y_{i}\parallel^{2})$

the second part is my question

prove that if $\displaystyle \{y_{1}..y_{n}\}$ is a set for which $\displaystyle \sum_{i=1}^{n}\parallel y_{i}\parallel^{2}<1 $then $\displaystyle \{y_{1}+u_{1,}..,y_{n}+u_{n}\}$ is independant?

how i tried to solve it:

a set is independant if for $\displaystyle k_{1}(y_{1}+u_{1})+..+k_{n}(y_{n}+u_{n})=0$the only way it to be zero is if k1=..=kn=0

i opened it and got $\displaystyle k_{1}y_{1}+..+k_{n}y_{n}=k_{1}u_{1}+..+k_{n}u_{n}$

so on both side we have equal vectors so is their norm

$\displaystyle (\sum_{i=1}^{n}|k_{i}|^{2})(\sum_{i=1}^{n}\paralle l y_{i}\parallel^{2})=(\sum_{i=1}^{n}|k_{i}|^{2})(\s um_{i=1}^{n}\parallel u_{i}\parallel^{2})

$what next how to prove that $\displaystyle k_{1}=..=k_{n}=0$

??