This repository has been archived by the owner on Jul 24, 2024. It is now read-only.
[Merged by Bors] - feat(analysis/inner_product_space/pi_L2): norms of basis vectors #19020
Conversation
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
eric-wieser
added
awaiting-review
eric-wieser
commented
May 15, 2023
Comment on lines
+659
to
+679
| @[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 |
There was a problem hiding this comment.
This one was an annoying case bash, but maybe that's unavoidable since both the norm and pi.single are defined by cases.
sgouezel
reviewed
May 17, 2023
Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr>
|
bors d+ |
|
✌️ eric-wieser can now approve this pull request. To approve and merge a pull request, simply reply with |
github-actions
bot
added
delegated
|
bors merge |
github-actions
bot
added
the
ready-to-merge
|
Pull request successfully merged into master. Build succeeded! The publicly hosted instance of bors-ng is deprecated and will go away soon. If you want to self-host your own instance, instructions are here. If you want to switch to GitHub's built-in merge queue, visit their help page. |
bors
bot
changed the title
Sign up for free
to subscribe to this conversation on GitHub.
Already have an account?
Sign in.
Add this suggestion to a batch that can be applied as a single commit.
This suggestion is invalid because no changes were made to the code.
Suggestions cannot be applied while the pull request is closed.
Suggestions cannot be applied while viewing a subset of changes.
Only one suggestion per line can be applied in a batch.
Add this suggestion to a batch that can be applied as a single commit.
Applying suggestions on deleted lines is not supported.
You must change the existing code in this line in order to create a valid suggestion.
Outdated suggestions cannot be applied.
This suggestion has been applied or marked resolved.
Suggestions cannot be applied from pending reviews.
Suggestions cannot be applied on multi-line comments.
Suggestions cannot be applied while the pull request is queued to merge.
Suggestion cannot be applied right now. Please check back later.
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.