= Coordinate functional representation of an operator into bounded indexed functions
{title2=$\|T\|=\sup_\gamma\|f_\gamma\|$}
A <bounded linear operator> $T:X\to\ell_\infty(\Gamma)$ is exactly a uniformly bounded family of <bounded linear functionals> $f_\gamma\in X^*$ through $(Tx)(\gamma)=f_\gamma(x)$. Evaluation shows $\|f_\gamma\|\le\|T\|$; conversely the family bound gives $\|Tx\|\le(\sup\|f_\gamma\|)\|x\|$, proving equality. Taking all functionals in the dual unit ball gives an <isometric embedding> by the <Hahn-Banach theorem>.
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