A variant of the Gram Schmidt process using transfinite recursion applied to a ( possibly uncountably ) infinite sequence of vectors ( v _ \ alpha ) _ { \ alpha yields a set of orthonormal vectors ( u _ \ alpha ) _ { \ alpha with \ kappa \ leq \ lambda such that for any \ alpha \ leq \ lambda, the completion of the span of \ lbrace u _ \ beta : \ beta is the same as that of \ lbrace v _ \ beta : \ beta.
