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[Merged by Bors] - feat(analysis/locally_convex): locally bounded implies continuous#16550
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[Merged by Bors] - feat(analysis/locally_convex): locally bounded implies continuous#16550
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| /-- If `E` is first countable, then every locally bounded linear map `E →ₛₗ[σ] F` is continuous. -/ | ||
| lemma linear_map.continuous_of_locally_bounded [uniform_add_group F] (f : E →ₛₗ[σ] F) |
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Is it interesting to turn this into an iff together with is_vonN_bounded.image (under suitable assumptions for σ)?
Other than that, LGTM.
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The topological vector spaces for which one has an iff is actually one equivalent definition of bornological spaces, so I don't want to put the iff version just here, because this lemma will be used to show that if the topology is first countable then the space is bornological.
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…6550) We prove that locally bounded linear maps are continuous provided the domain is first countable. In the literature this is usually stated with pseudometrizable, but first countable is equivalent.
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…sis.normed_space.is_R_or_C` later (#16758) We don't want to import `analysis/normed_space/is_R_or_C` in `analysis/locally_convex/bounded` because it will create a cycle when defining the strong topology on `continuous_linear_map`, because we will have to (transitively) import `analysis/locally_convex/bounded` in `analysis/normed_space/operator_norm`. So I moved the material of #16550 in a new file `analysis/locally_convex/continuous_of_bounded`
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We prove that locally bounded linear maps are continuous provided the domain is first countable. In the literature this is usually stated with pseudometrizable, but first countable is equivalent.
I am unsure about the bundling, see zulip. Either this or the previous lemma have to be changed.