Product of $L_2$ norm of vectors

Question

Is the $\sum \Vert b_k\Vert_2^2 \le\ge= \sum \Vert b_k\Vert_2^2 \Vert a_k\Vert_2^2$ ?

where $b_k$ is a column vector and $a_k$ is a highly sparse row vector.

Answer 1

You can't say anything... Take $a_k = (s,0,\ldots,0)$ with $s\in \{10^{-132},1,10^{345}\}$ for every $k$. Then you will face all three situations ($\leq,\geq$ and $=$).