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Riesz's lemma #

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.

theorem riesz_lemma {𝕜 : Type u_1} [normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {F : subspace 𝕜 E} (hFc : is_closed F) (hF : ∃ (x : E), x F) {r : } (hr : r < 1) :
∃ (x₀ : E), x₀ F ∀ (y : E), y Fr * x₀ x₀ - y

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.