Documentation

Mathlib.Analysis.Normed.Group.HomCompletion

Completion of normed group homs #

Given two (semi) normed groups G and H and a normed group hom f : NormedAddGroupHom G H, we build and study a normed group hom f.completion : NormedAddGroupHom (completion G) (completion H) such that the diagram

                   f
     G       ----------->     H

     |                        |
     |                        |
     |                        |
     V                        V

completion G -----------> completion H
            f.completion

commutes. The map itself comes from the general theory of completion of uniform spaces, but here we want a normed group hom, study its operator norm and kernel.

The vertical maps in the above diagrams are also normed group homs constructed in this file.

Main definitions and results: #

The normed group hom induced between completions.

Equations
Instances For
    theorem NormedAddGroupHom.completion_coe {G : Type u_1} [SeminormedAddCommGroup G] {H : Type u_2} [SeminormedAddCommGroup H] (f : NormedAddGroupHom G H) (g : G) :
    f.completion (G g) = H (f g)
    @[simp]
    theorem NormedAddGroupHom.completion_coe' {G : Type u_1} [SeminormedAddCommGroup G] {H : Type u_2} [SeminormedAddCommGroup H] (f : NormedAddGroupHom G H) (g : G) :
    UniformSpace.Completion.map (f) (G g) = H (f g)
    @[simp]
    theorem normedAddGroupHomCompletionHom_apply {G : Type u_1} [SeminormedAddCommGroup G] {H : Type u_2} [SeminormedAddCommGroup H] (f : NormedAddGroupHom G H) :
    normedAddGroupHomCompletionHom f = f.completion

    Completion of normed group homs as a normed group hom.

    Equations
    • normedAddGroupHomCompletionHom = { toFun := NormedAddGroupHom.completion, map_zero' := , map_add' := }
    Instances For
      theorem NormedAddGroupHom.completion_comp {G : Type u_1} [SeminormedAddCommGroup G] {H : Type u_2} [SeminormedAddCommGroup H] {K : Type u_3} [SeminormedAddCommGroup K] (f : NormedAddGroupHom G H) (g : NormedAddGroupHom H K) :
      g.completion.comp f.completion = (g.comp f).completion
      theorem NormedAddGroupHom.completion_neg {G : Type u_1} [SeminormedAddCommGroup G] {H : Type u_2} [SeminormedAddCommGroup H] (f : NormedAddGroupHom G H) :
      (-f).completion = -f.completion
      theorem NormedAddGroupHom.completion_add {G : Type u_1} [SeminormedAddCommGroup G] {H : Type u_2} [SeminormedAddCommGroup H] (f : NormedAddGroupHom G H) (g : NormedAddGroupHom G H) :
      (f + g).completion = f.completion + g.completion
      theorem NormedAddGroupHom.completion_sub {G : Type u_1} [SeminormedAddCommGroup G] {H : Type u_2} [SeminormedAddCommGroup H] (f : NormedAddGroupHom G H) (g : NormedAddGroupHom G H) :
      (f - g).completion = f.completion - g.completion
      @[simp]
      theorem NormedAddCommGroup.toCompl_apply {G : Type u_1} [SeminormedAddCommGroup G] :
      ∀ (a : G), NormedAddCommGroup.toCompl a = G a

      The map from a normed group to its completion, as a normed group hom.

      Equations
      • NormedAddCommGroup.toCompl = { toFun := G, map_add' := , bound' := }
      Instances For
        theorem NormedAddCommGroup.norm_toCompl {G : Type u_1} [SeminormedAddCommGroup G] (x : G) :
        NormedAddCommGroup.toCompl x = x
        theorem NormedAddCommGroup.denseRange_toCompl {G : Type u_1} [SeminormedAddCommGroup G] :
        DenseRange NormedAddCommGroup.toCompl
        @[simp]
        theorem NormedAddGroupHom.completion_toCompl {G : Type u_1} [SeminormedAddCommGroup G] {H : Type u_2} [SeminormedAddCommGroup H] (f : NormedAddGroupHom G H) :
        f.completion.comp NormedAddCommGroup.toCompl = NormedAddCommGroup.toCompl.comp f
        theorem NormedAddGroupHom.ker_le_ker_completion {G : Type u_1} [SeminormedAddCommGroup G] {H : Type u_2} [SeminormedAddCommGroup H] (f : NormedAddGroupHom G H) :
        (NormedAddCommGroup.toCompl.comp (NormedAddGroupHom.incl f.ker)).range f.completion.ker
        theorem NormedAddGroupHom.ker_completion {G : Type u_1} [SeminormedAddCommGroup G] {H : Type u_2} [SeminormedAddCommGroup H] {f : NormedAddGroupHom G H} {C : } (h : f.SurjectiveOnWith f.range C) :
        f.completion.ker = closure (NormedAddCommGroup.toCompl.comp (NormedAddGroupHom.incl f.ker)).range

        If H is complete, the extension of f : NormedAddGroupHom G H to a NormedAddGroupHom (completion G) H.

        Equations
        Instances For
          theorem NormedAddGroupHom.extension_def {G : Type u_1} [SeminormedAddCommGroup G] {H : Type u_2} [SeminormedAddCommGroup H] [T0Space H] [CompleteSpace H] (f : NormedAddGroupHom G H) (v : G) :
          f.extension (G v) = UniformSpace.Completion.extension (f) (G v)
          @[simp]
          theorem NormedAddGroupHom.extension_coe {G : Type u_1} [SeminormedAddCommGroup G] {H : Type u_2} [SeminormedAddCommGroup H] [T0Space H] [CompleteSpace H] (f : NormedAddGroupHom G H) (v : G) :
          f.extension (G v) = f v
          theorem NormedAddGroupHom.extension_unique {G : Type u_1} [SeminormedAddCommGroup G] {H : Type u_2} [SeminormedAddCommGroup H] [T0Space H] [CompleteSpace H] (f : NormedAddGroupHom G H) {g : NormedAddGroupHom (UniformSpace.Completion G) H} (hg : ∀ (v : G), f v = g (G v)) :
          f.extension = g