Minimal total family in a normed space ($\overline{\mathrm{span}}\{x_i:i\in I\}=X,x_i\notin \overline{\mathrm{span}}\{x_j:j\ne i\}$)

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 $M_i=\overline{\mathrm{span}}\{x_j:j\ne i\}$ and $d_i=\mathrm{dist}(x_i,M_i)>0$. The functional $m+t{x}_i\mapsto t$ on $M_i+\mathbb{R}{x}_i$ 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 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.