English
The NormedAddCommGroup can be equipped with a uniform structure equal to a given one and the bornology matches.
Русский
Ngомплектуется униформная структура равной заданной и борология совпадает.
LaTeX
$$$$\mathcal{U}_{new}=\mathcal{U}_{given},\quad \text{Bornology}_{new}=\text{Bornology}_{given}.$$$$
Lean4
/-- Produces a `NormedAddCommGroup E` instance from a `NormedSpace.Core` on a type
that already has an existing uniform space structure. This requires a proof that the uniformity
induced by the norm is equal to the preexisting uniformity. See note [reducible non-instances]. -/
abbrev ofCoreReplaceUniformity [U : UniformSpace E] (core : NormedSpace.Core 𝕜 E)
(H : 𝓤[U] = 𝓤[PseudoEMetricSpace.toUniformSpace (self := PseudoEMetricSpace.ofSeminormedSpaceCore core.toCore)]) :
NormedAddCommGroup E :=
{ SeminormedAddCommGroup.ofCoreReplaceUniformity core.toCore H with
eq_of_dist_eq_zero := by
intro x y h
rw [← sub_eq_zero, ← core.norm_eq_zero_iff]
exact h }
