feat(Analysis/LocallyConvex/Polar): Bipolar theorem #26345
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Co-authored-by: Jireh Loreaux <loreaujy@gmail.com>
This reverts commit f8148c2.
Co-authored-by: Jireh Loreaux <loreaujy@gmail.com>
Co-authored-by: Filippo A. E. Nuccio <filippo.nuccio@univ-st-etienne.fr>
Co-authored-by: Filippo A. E. Nuccio <filippo.nuccio@univ-st-etienne.fr>
Co-authored-by: Filippo A. E. Nuccio <filippo.nuccio@univ-st-etienne.fr>
Co-authored-by: Filippo A. E. Nuccio <filippo.nuccio@univ-st-etienne.fr>
Co-authored-by: Filippo A. E. Nuccio <filippo.nuccio@univ-st-etienne.fr>
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@mans0954 I let you check all my comments, then please ping me so that I can come back for another round of review. |
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… a single ring of scalars (#29342) Currently the definition of Absolutely Convex in Mathlib is a little unexpected: ``` def AbsConvex (s : Set E) : Prop := Balanced 𝕜 s ∧ Convex ℝ s ``` At the time this definition was formulated, Mathlib's definition of `Convex` required the scalars to be an `OrderedSemiring` whereas the definition of `Balanced` required the scalars to be a `SeminormedRing`. Mathlib didn't have a concept of a semi-normed ordered ring, so a set was defined as `AbsConvex` if it is balanced over a `SeminormedRing` `𝕜` and convex over `ℝ`. Recently the requirements for the definition of `Convex` have been relaxed (#24392, #20595) so it is now possible to use a single scalar ring in common with the literature. Previous discussion: - #17029 (comment) - #26345 (comment)
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… a single ring of scalars (leanprover-community#29342) Currently the definition of Absolutely Convex in Mathlib is a little unexpected: ``` def AbsConvex (s : Set E) : Prop := Balanced 𝕜 s ∧ Convex ℝ s ``` At the time this definition was formulated, Mathlib's definition of `Convex` required the scalars to be an `OrderedSemiring` whereas the definition of `Balanced` required the scalars to be a `SeminormedRing`. Mathlib didn't have a concept of a semi-normed ordered ring, so a set was defined as `AbsConvex` if it is balanced over a `SeminormedRing` `𝕜` and convex over `ℝ`. Recently the requirements for the definition of `Convex` have been relaxed (leanprover-community#24392, leanprover-community#20595) so it is now possible to use a single scalar ring in common with the literature. Previous discussion: - leanprover-community#17029 (comment) - leanprover-community#26345 (comment)
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… a single ring of scalars (leanprover-community#29342) Currently the definition of Absolutely Convex in Mathlib is a little unexpected: ``` def AbsConvex (s : Set E) : Prop := Balanced 𝕜 s ∧ Convex ℝ s ``` At the time this definition was formulated, Mathlib's definition of `Convex` required the scalars to be an `OrderedSemiring` whereas the definition of `Balanced` required the scalars to be a `SeminormedRing`. Mathlib didn't have a concept of a semi-normed ordered ring, so a set was defined as `AbsConvex` if it is balanced over a `SeminormedRing` `𝕜` and convex over `ℝ`. Recently the requirements for the definition of `Convex` have been relaxed (leanprover-community#24392, leanprover-community#20595) so it is now possible to use a single scalar ring in common with the literature. Previous discussion: - leanprover-community#17029 (comment) - leanprover-community#26345 (comment)
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@mans0954 Are you planning to keep on working on this PR? |
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The bipolar theorem states that the polar of the polar of a set
sis equal to the closed absolutely convex hull ofs.The argument here follows Conway, Chapter V. 1.8.
This PR continues the work from #20843.
Original PR: #20843