[Merged by Bors] - feat(analysis/normed_space/SemiNormedGroup): has_cokernels #7628
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From LTE. Co-authored-by: Riccardo Brasca <riccardo.brasca@gmail.com>
Thanks Benjamin! Co-authored-by: Benjamin Davidson <68528197+benjamindavidson@users.noreply.github.com>
…NormedGroup_cokernel
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Co-authored-by: Riccardo Brasca <riccardo.brasca@gmail.com>
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| lemma lift_bound_by {N : Type*} [semi_normed_group N] (S : add_subgroup M) | ||
| (f : normed_group_hom M N) (hf : ∀ s ∈ S, f s = 0) | ||
| {c : ℝ≥0} (fb : f.bound_by c) : | ||
| (lift S f hf).bound_by c := |
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I realize that I was just being completely ignorant with this bound_by definition. Would you mind phrasing this in terms of the operator norm?
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# Cokernels in SemiNormedGroup₁ and SemiNormedGroup We show that `SemiNormedGroup₁` has cokernels (for which of course the `cokernel.π f` maps are norm non-increasing), as well as the easier result that `SemiNormedGroup` has cokernels. So far, I don't see a way to state nicely what we really want: `SemiNormedGroup` has cokernels, and `cokernel.π f` is norm non-increasing. The problem is that the limits API doesn't promise you any particular model of the cokernel, and in `SemiNormedGroup` one can always take a cokernel and rescale its norm (and hence making `cokernel.π f` arbitrarily large in norm), obtaining another categorical cokernel. Co-authored-by: Patrick Massot <patrickmassot@free.fr> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>
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Cokernels in SemiNormedGroup₁ and SemiNormedGroup
We show that
SemiNormedGroup₁has cokernels(for which of course the
cokernel.π fmaps are norm non-increasing),as well as the easier result that
SemiNormedGrouphas cokernels.So far, I don't see a way to state nicely what we really want:
SemiNormedGrouphas cokernels, andcokernel.π fis norm non-increasing.The problem is that the limits API doesn't promise you any particular model of the cokernel,
and in
SemiNormedGroupone can always take a cokernel and rescale its norm(and hence making
cokernel.π farbitrarily large in norm), obtaining another categorical cokernel.