[Merged by Bors] - feat(Analysis/LocallyConvex/WithSeminorms): characterize continuous seminorms #5501
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…eminorms (#5501) This shows that, if the topology of `E` is defined by some family of seminorms `p`, then a seminorm `q` is continuous iff `∃ s : Finset ι, ∃ C : ℝ≥0, C ≠ 0 ∧ q ≤ C • s.sup p`. Via [Seminorm.continuous_iff_continuous_comp](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/LocallyConvex/WithSeminorms.html#Seminorm.continuous_iff_continuous_comp) this gives the converse of [Seminorm.continuous_from_bounded](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/LocallyConvex/WithSeminorms.html#Seminorm.continuous_from_bounded) and hence a characterization of continuous linear maps between such spaces. To do that, we restate all of the "bound of shell" lemmas in terms of seminorms, which needs changing some imports, but I've checked the current state of the port and this should not cause too much trouble since most of the touched files are already ported so we can changes the imports in mathlib4 too. The `WithSeminorms` file needs a naming/dot notation refactor at some point, because the naming scheme is neither predictable nor convenient to use, but this PR is already large enough.
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…eminorms (#5501) This shows that, if the topology of `E` is defined by some family of seminorms `p`, then a seminorm `q` is continuous iff `∃ s : Finset ι, ∃ C : ℝ≥0, C ≠ 0 ∧ q ≤ C • s.sup p`. Via [Seminorm.continuous_iff_continuous_comp](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/LocallyConvex/WithSeminorms.html#Seminorm.continuous_iff_continuous_comp) this gives the converse of [Seminorm.continuous_from_bounded](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/LocallyConvex/WithSeminorms.html#Seminorm.continuous_from_bounded) and hence a characterization of continuous linear maps between such spaces. To do that, we restate all of the "bound of shell" lemmas in terms of seminorms, which needs changing some imports, but I've checked the current state of the port and this should not cause too much trouble since most of the touched files are already ported so we can changes the imports in mathlib4 too. The `WithSeminorms` file needs a naming/dot notation refactor at some point, because the naming scheme is neither predictable nor convenient to use, but this PR is already large enough.
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…eminorms (#5501) This shows that, if the topology of `E` is defined by some family of seminorms `p`, then a seminorm `q` is continuous iff `∃ s : Finset ι, ∃ C : ℝ≥0, C ≠ 0 ∧ q ≤ C • s.sup p`. Via [Seminorm.continuous_iff_continuous_comp](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/LocallyConvex/WithSeminorms.html#Seminorm.continuous_iff_continuous_comp) this gives the converse of [Seminorm.continuous_from_bounded](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Analysis/LocallyConvex/WithSeminorms.html#Seminorm.continuous_from_bounded) and hence a characterization of continuous linear maps between such spaces. To do that, we restate all of the "bound of shell" lemmas in terms of seminorms, which needs changing some imports, but I've checked the current state of the port and this should not cause too much trouble since most of the touched files are already ported so we can changes the imports in mathlib4 too. The `WithSeminorms` file needs a naming/dot notation refactor at some point, because the naming scheme is neither predictable nor convenient to use, but this PR is already large enough.
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This shows that, if the topology of
Eis defined by some family of seminormsp, then a seminormqis continuous iff∃ s : Finset ι, ∃ C : ℝ≥0, C ≠ 0 ∧ q ≤ C • s.sup p. Via Seminorm.continuous_iff_continuous_comp this gives the converse of Seminorm.continuous_from_bounded and hence a characterization of continuous linear maps between such spaces.To do that, we restate all of the "bound of shell" lemmas in terms of seminorms, which needs changing some imports, but I've checked the current state of the port and this should not cause too much trouble since most of the touched files are already ported so we can changes the imports in mathlib4 too.
The
WithSeminormsfile needs a naming/dot notation refactor at some point, because the naming scheme is neither predictable nor convenient to use, but this PR is already large enough.This is a manual port of leanprover-community/mathlib#17298 which was already
maintainer merge-d andbors delegate-d, but I thought it would be easier to just make a new Mathlib4 PR.There's one thing I'm not sure about: I moved some lemmas to other files (and I moved the
#aligns with them), will that be a problem for mathport? If so we should also merge the Mathlib3 PR, otherwise I can just close it.