 A variant of the Gram–Schmidt process using transfinite recursion applied to a (possibly uncountably) infinite sequence of vectors formula_8 yields a set of orthonormal vectors formula_9 with formula_10 such that for any formula_11, the completion of the span of formula_12 is the same as that of formula_13.In particular, when applied to a (algebraic) basis of a Hilbert space (or, more generally, a basis of any dense subspace), it yields a (functional-analytic) orthonormal basis.Note that in the general case often the strict inequality formula_14 holds, even if the starting set was linearly independent, and the span of formula_9 need not be a subspace of the span of formula_8 (rather, it's a subspace of its completion).