Prove inner-product inequality $\Bigl (\sum_{i=1}^k \langle u\;,\;e_i \rangle \langle v\;,\;e_i \rangle \Bigr)^2≤ \Vert u \Vert^2 \, \Vert v \Vert^2$
Proving $\sum\limits_{i=1}^k | \langle x,v_i \rangle \langle y,v_i\rangle| \leq \|x\|\cdot \|y\|$