Reverse of Cauchy-Schwarz for two vectors

Question

If $\|x\|_2\leq g\|y\|_2$, then according to Cauchy-Schuarz, the maximum of $ \langle x,y \rangle$ would be $ g\|y\|_2^2$

$$ \langle x,y \rangle \leq g\|y\|_2^2 $$ Can we prove the reverse, i.e., if $\langle x,y \rangle \leq g\|y\|_2^2$, then $$ \|x\|_2\leq g\|y\|_2 $$

Answer 1

No -- if $x$ is orthogonal to $y$, we can make $x$ as big as we want but still have inner product $0$