# mathlib3documentation

analysis.normed.group.SemiNormedGroup.completion

# Completions of normed groups #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

This file contains an API for completions of seminormed groups (basic facts about objects and morphisms).

## Main definitions #

• SemiNormedGroup.Completion : SemiNormedGroup ⥤ SemiNormedGroup : the completion of a seminormed group (defined as a functor on SemiNormedGroup to itself).
• SemiNormedGroup.Completion.lift (f : V ⟶ W) : (Completion.obj V ⟶ W) : a normed group hom from V to complete W extends ("lifts") to a seminormed group hom from the completion of V to W.

## Projects #

1. Construct the category of complete seminormed groups, say CompleteSemiNormedGroup and promote the Completion functor below to a functor landing in this category.
2. Prove that the functor Completion : SemiNormedGroup ⥤ CompleteSemiNormedGroup is left adjoint to the forgetful functor.
noncomputable def SemiNormedGroup.Completion  :

The completion of a seminormed group, as an endofunctor on SemiNormedGroup.

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Instances for SemiNormedGroup.Completion
@[simp]
theorem SemiNormedGroup.Completion_map (V W : SemiNormedGroup) (f : V W) :
@[simp]
@[protected, instance]

The canonical morphism from a seminormed group V to its completion.

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noncomputable def SemiNormedGroup.Completion.map_hom (V W : SemiNormedGroup) :
(V W) →+

Given a normed group hom V ⟶ W, this defines the associated morphism from the completion of V to the completion of W. The difference from the definition obtained from the functoriality of completion is in that the map sending a morphism f to the associated morphism of completions is itself additive.

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@[protected, instance]
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@[protected, instance]
noncomputable def SemiNormedGroup.Completion.lift {V W : SemiNormedGroup} (f : V W) :

Given a normed group hom f : V → W with W complete, this provides a lift of f to the completion of V. The lemmas lift_unique and lift_comp_incl provide the api for the universal property of the completion.

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