  A fundamental inequality for Hilbert spaces is the $\ell_1-\ell_2$-norm
inequality which gives that for any $x \in H^n$, $\|x\|_1\le \sqrt{n}\|x\|_2.$
But this is a strict inequality for all but vectors with constant modulus for
their coefficients. We will give a trivial method to compute, for each x, the
constant $c$ for which $\|x\|_1=c\sqrt{n}\|x\|_2.$ Since this inequality is one
of the most used results in Hilbert space theory, we believe this will have
unlimited applications in the field. We will also show some variations of this
result.
