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[Merged by Bors] - feat(analysis/inner_product_space/pi_L2): norms of basis vectors - #19020
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[Merged by Bors] - feat(analysis/inner_product_space/pi_L2): norms of basis vectors#19020eric-wieser wants to merge 2 commits into
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| @[simp] | ||
| lemma nnnorm_equiv_symm_single (i : ι) (b : β i) : | ||
| ‖(pi_Lp.equiv p β).symm (pi.single i b)‖₊ = ‖b‖₊ := | ||
| begin | ||
| haveI : nonempty ι := ⟨i⟩, | ||
| unfreezingI { induction p using with_top.rec_top_coe }, | ||
| { simp_rw [nnnorm_eq_csupr, equiv_symm_apply], | ||
| refine csupr_eq_of_forall_le_of_forall_lt_exists_gt (λ j, _) (λ n hn, ⟨i, hn.trans_eq _⟩), | ||
| { obtain rfl | hij := decidable.eq_or_ne i j, | ||
| { rw pi.single_eq_same }, | ||
| { rw [pi.single_eq_of_ne' hij, nnnorm_zero], | ||
| exact zero_le _ } }, | ||
| { rw pi.single_eq_same } }, | ||
| { have hp0 : (p : ℝ) ≠ 0, | ||
| { exact_mod_cast (zero_lt_one.trans_le $ fact.out (1 ≤ (p : ℝ≥0∞))).ne' }, | ||
| rw [nnnorm_eq_sum ennreal.coe_ne_top, ennreal.coe_to_real, fintype.sum_eq_single i, | ||
| equiv_symm_apply, pi.single_eq_same, ←nnreal.rpow_mul, one_div, mul_inv_cancel hp0, | ||
| nnreal.rpow_one], | ||
| intros j hij, | ||
| rw [equiv_symm_apply, pi.single_eq_of_ne hij, nnnorm_zero, nnreal.zero_rpow hp0] }, | ||
| end |
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This one was an annoying case bash, but maybe that's unavoidable since both the norm and pi.single are defined by cases.
sgouezel
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May 17, 2023
Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr>
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This adds
‖euclidean_space.single i (a : 𝕜)‖ = ‖a‖and other similar results.They hold more generally for
pi_Lp, so they are proven there first.The statement of
linear_isometry_equiv.pi_Lp_congr_left_singlehas also been corrected to include the missing typecast.This was originally motivated by #19013, though I think I don't actually need it there after all. I think the results are still worth having though.