[Merged by Bors] - feat(analysis/normed_space/basic): add norm_algebra_map_nnreal
#16709
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
j-loreaux
added
awaiting-review
|
bors d+ Feel free to merge once CI passes. |
|
✌️ j-loreaux can now approve this pull request. To approve and merge a pull request, simply reply with |
github-actions
bot
added
delegated
This was referenced Sep 29, 2022
|
bors r+ |
bors bot
pushed a commit
that referenced
this pull request
Sep 30, 2022
) This adds `simp` lemmas saying that `∥algebra_map ℝ≥0 𝕜 x∥ = x` and similarly for `∥⬝∥₊` whenever `𝕜` is a normed `ℝ`-algebra and satisfies `norm_one_class`. These are needed separately from `norm_algebra_map'` and `nnnorm_algebra_map'` because `𝕜` cannot be a normed `ℝ≥0`-algebra for the simple reason that `ℝ≥0` is not a normed field.
|
Pull request successfully merged into master. Build succeeded: |
bors
bot
changed the title
norm_algebra_map_nnrealnorm_algebra_map_nnreal
bors bot
pushed a commit
that referenced
this pull request
Oct 7, 2022
…spondence between closed ideals in `C(X, 𝕜)` and open sets in `X` (#16677) For a topological ring `R` and a topological space `X` there is a Galois connection between `ideal C(X, R)` and `set X` given by sending each `I : ideal C(X, R)` to `{x : X | ∀ f ∈ I, f x = 0}ᶜ` and mapping `s : set X` to the ideal with carrier `{f : C(X, R) | ∀ x ∈ sᶜ, f x = 0}`, and we call these maps `continuous_map.set_of_ideal` and `continuous_map.ideal_of_set`. As long as `R` is Hausdorff, `continuous_map.set_of_ideal I` is open, and if, in addition, `X` is locally compact, then `continuous_map.set_of_ideal s` is closed. When `R = 𝕜` with `is_R_or_C 𝕜` and `X` is compact Hausdorff, then this Galois connection can be improved to a true Galois correspondence (i.e., order isomorphism) between the type `opens X` and the subtype of closed ideals of `C(X, R)`. As a consequence, the maximal ideals correspond precisely to (complements of) singletons, but this fact will appear in a separate PR. - [x] depends on: #16664 - [x] depends on: #16709 - [x] depends on: #16713 - [x] depends on: #16714
bors bot
pushed a commit
that referenced
this pull request
Oct 24, 2022
…to (complements of) singletons (#16719) This establishes the correspondence between maximal ideals of `C(X, 𝕜)` (where `X` is compact Hausdorff and `is_R_or_C 𝕜`) and the complements of singletons in `X`. This allows for the proof that the natural map from `X` to `character_space 𝕜 C(X, 𝕜)` is a homeomorphism. - [x] depends on: #16709 - [x] depends on: #16713 - [x] depends on: #16714 - [x] depends on: #16677 - [x] depends on: #16663 - [x] depends on: #16721 - [x] depends on: #16722 - [x] depends on: #16801
Sign up for free
to join this conversation on GitHub.
Already have an account?
Sign in to comment
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
simplemmas saying that∥algebra_map ℝ≥0 𝕜 x∥ = xand similarly for∥⬝∥₊whenever𝕜is a normedℝ-algebra and satisfiesnorm_one_class. These are needed separately fromnorm_algebra_map'andnnnorm_algebra_map'because𝕜cannot be a normedℝ≥0-algebra for the simple reason thatℝ≥0is not a normed field.