= Norming functional
{title2=$\|\ell\|=1,\quad\ell(v)=\|v\|$}
For a nonzero <vector> in a real or complex <normed vector space>, a <norming functional> is a <continuous linear functional> satisfying the displayed conditions. The <Hahn-Banach theorem> extends the functional sending $av$ to $a\|v\|$ on its one-dimensional span. Consequently $\|w\|\geq\operatorname{Re}\ell(w)$, with equality at $v$, giving a supporting linear <lower bound> for the <norm>. At zero the zero functional can be used as a supporting functional.
Back to article page
