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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.

A biorthogonal system is a family $(x_i,f_i{)}_{i\in I}$ with $x_i\in X$, $f_i\in X^{\ast }$, satisfyingThus each continuous coordinate functional takes value one on its own vector and zero on all others. The vectors need not be orthogonal in any inner product, and totality of their linear span is an additional condition.

For a countable minimal total family in a normed space in a Banach space, let $(x_n,f_n)$ be its unique biorthogonal system and let $P_{n}x={\sum }_{j\le n}{f}_j(x)x_j$. Uniform boundedness of these projections implies $P_{n}x\to x$ for every $x$: approximate $x$ by a finite linear combination $y$, eventually fixed by $P_n$, and bound the error by $(1+{\sup }_n\Vert P_n\Vert )\Vert x-y\Vert$. Conversely, if every vector has a norm-convergent expansion in this family, applying each continuous $f_j$ shows that its coefficients must be $f_j(x)$. Thus $P_{n}x\to x$, so the projections are pointwise bounded. The Uniform boundedness principle gives their uniform operator-norm bound. This proves the criterion and uniqueness of all expansions.