= Minimal total family in a normed space
{title2=$\overline{\operatorname{span}}\{x_i:i\in I\}=X,\quad x_i\notin\overline{\operatorname{span}}\{x_j:j\ne i\}$}
A total family has dense <linear span>; it is minimal when no <vector> lies in the closed span of all the others. Equivalently, no proper subfamily remains total. Such a family has unique bounded coordinate functionals making a <biorthogonal system>. Indeed put $M_i=\overline{\operatorname{span}}\{x_j:j\ne i\}$ and $d_i=\operatorname{dist}(x_i,M_i)>0$. The functional $m+tx_i\mapsto t$ on $M_i+\mathbb Rx_i$ is bounded by $1/d_i$, so the <Hahn-Banach theorem> extends it to $X$. Its <norm> is exactly $1/d_i$, by the distance infimum. Uniqueness follows because its values on the total family determine it on a dense <vector subspace>[subspace]. For a countable family, minimality and totality alone do not ensure norm-convergent expansions; the <uniform partial-sum criterion for a Schauder basis> supplies the additional condition.
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