[Merged by Bors] - feat(analysis/normed_space/finite_dimension): Riesz theorem on compact unit ball and finite dimension #9147
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sgouezel
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Sep 11, 2021
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| /-- A sequence of points in an infinite-dimensional space, which are all bounded by `R` and at | ||
| distance at least `1`. Use `exists_seq_norm_le_le_norm_sub` instead. -/ | ||
| noncomputable def exists_seq_norm_le_one_le_norm_sub_aux |
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| noncomputable def exists_seq_norm_le_one_le_norm_sub_aux | |
| noncomputable def seq_norm_le_one_le_norm_sub_aux |
Should it be private?
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I think Floris told me that he hated private declarations because they can't be printed or checked, so I don't mark anything as private any more.
| `exists_seq_norm_le_one_le_norm_sub`. -/ | ||
| theorem exists_seq_norm_le_one_le_norm_sub' {c : 𝕜} (hc : 1 < ∥c∥) {R : ℝ} (hR : ∥c∥ < R) | ||
| (h : ¬ (finite_dimensional 𝕜 E)) : | ||
| ∃ f : ℕ → E, (∀ n, ∥f n∥ ≤ R) ∧ (∀ m n, m ≠ n → 1 ≤ ∥f m - f n∥) := |
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I saw this kind of induction more than once. Should we have a lemma like this one?
lemma nat.exists_fun_forall_lt_imp {α : Type*} (p : α → Prop) (r : α → α → Prop)
(H : ∀ (n : ℕ) (f : ℕ → α), ∃ y, p y ∧ ∀ k < n, r (f k) y) :
∃ f : ℕ → α, (∀ n, p n) ∧ ∀ m n, m < n → r m n := sorryThere was a problem hiding this comment.
BTW, H can assume that f satisfies ∀ k < n, p (f k) and ∀ k l, k < l → l < n → r (f k) (f l).
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I have added a version expressed in terms of finsets: if for any finset s one can find a point y with r x y for all points in the finset, then one can find a sequence with r (f m) (f n) whenever m < n.
| `exists_seq_norm_le_one_le_norm_sub`. -/ | ||
| theorem exists_seq_norm_le_one_le_norm_sub' {c : 𝕜} (hc : 1 < ∥c∥) {R : ℝ} (hR : ∥c∥ < R) | ||
| (h : ¬ (finite_dimensional 𝕜 E)) : | ||
| ∃ f : ℕ → E, (∀ n, ∥f n∥ ≤ R) ∧ (∀ m n, m ≠ n → 1 ≤ ∥f m - f n∥) := |
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BTW, H can assume that f satisfies ∀ k < n, p (f k) and ∀ k l, k < l → l < n → r (f k) (f l).
Co-authored-by: Yury G. Kudryashov <urkud@urkud.name>
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Thanks! 🎉 |
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