Let 
be endowed with the usual product and the usual norm. If 
then we define 
. Prove that
![\[\left \| v \right \|^2 \left \| w \right \|^2 \geq \left ( v \cdot w \right )^2 + \frac{\left ( \left \| v \right \| \left | \sum w \right | - \left \| w \right \| \left |\sum v \right | \right )^2}{n}\]](https://www.math.tolaso.com.gr/wp-content/ql-cache/quicklatex.com-ce21dd6ae985525780de3af675d32523_l3.png "Rendered by QuickLaTeX.com")
Solution ( Robert Tauraso )
We will show the more general inequality
![\[\left ( \left \| v \right \|^2 \left \| w \right \|^2 - \left ( v \cdot w \right )^2 \right ) \left \| u \right \|^2 \geq \left \| \left ( w, u \right ) v - \left ( v, u \right ) w \right \|^2\]](https://www.math.tolaso.com.gr/wp-content/ql-cache/quicklatex.com-3f551aa275d780afd49b654a425b9561_l3.png "Rendered by QuickLaTeX.com")
where 
. Taking 
we get the requested inequality. If 
are linearly dependent then 
the inequality holds. We assume now that 
and 
are linearly independent. Then 
where 
and 
, 
. Moreover,
![\[\left\{\begin{matrix} \left ( v, u \right ) & = & \alpha \left \| v \right \|^2 + \beta \left ( v, w \right ) \\\\ \left ( w, u \right ) & = & \alpha \left ( v, w \right ) + \beta \left \| w \right \|^2 \end{matrix}\right.\]](https://www.math.tolaso.com.gr/wp-content/ql-cache/quicklatex.com-077b0d4d203169b64ba6c4ed310e066f_l3.png "Rendered by QuickLaTeX.com")
and by solving the linear system we find
![\[\alpha = \frac{\left ( v, u \right ) \left \| w \right \|^2 - \left ( w, u \right ) \left ( v, w \right )}{\left \| v \right \|^2 \left \| w \right \|^2 - \left ( v, w \right )^2} \quad , \quad \beta = \frac{\left ( w, u \right )\left \| u \right \|^2 - \left ( v, u \right )\left ( v, w \right )}{\left \| v \right \|^2 \left \| w \right \|^2 - \left ( v, w \right )^2}\]](https://www.math.tolaso.com.gr/wp-content/ql-cache/quicklatex.com-4bcd216915a2785cf3ed7aba9efe8975_l3.png "Rendered by QuickLaTeX.com")
Hence,
