Riesz's lemma, stated for a normed space over a normed field: for any closed proper subspace F of E, there is a nonzero x such that ∥x - F∥ is at least r * ∥x∥ for any r < 1.
Riesz's lemma, which usually states that it is possible to find a vector with norm 1 whose distance to a closed proper subspace is arbitrarily close to 1. The statement here is in terms of multiples of norms, since in general the existence of an element of norm exactly 1 is not guaranteed.