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Mathlib.Algebra.Order.Hom.Normed

Constructing (semi)normed groups from (semi)normed homs #

This file defines constructions that upgrade (Comm)Group to (Semi)Normed(Comm)Group using a Group(Semi)normClass when the codomain is the reals.

See Mathlib.Algebra.Order.Hom.Ultra for further upgrades to nonarchimedean normed groups.

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@[reducible, inline]
abbrev GroupSeminormClass.toSeminormedGroup {F : Type u_1} {α : Type u_2} [FunLike F α ℝ] [Group α] [GroupSeminormClass F α ℝ] (f : F) :
SeminormedGroup α

Constructs a SeminormedGroup structure from a GroupSeminormClass on a Group.

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    @[reducible, inline]
    abbrev AddGroupSeminormClass.toSeminormedAddGroup {F : Type u_1} {α : Type u_2} [FunLike F α ℝ] [AddGroup α] [AddGroupSeminormClass F α ℝ] (f : F) :
    SeminormedAddGroup α

    Constructs a SeminormedAddGroup structure from an AddGroupSeminormClass on an AddGroup.

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      theorem GroupSeminormClass.toSeminormedGroup_norm_eq {F : Type u_1} {α : Type u_2} [FunLike F α ℝ] [Group α] [GroupSeminormClass F α ℝ] (f : F) (x : α) :
      ‖x‖ = f x
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      theorem AddGroupSeminormClass.toSeminormedAddGroup_norm_eq {F : Type u_1} {α : Type u_2} [FunLike F α ℝ] [AddGroup α] [AddGroupSeminormClass F α ℝ] (f : F) (x : α) :
      ‖x‖ = f x
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      @[reducible, inline]
      abbrev GroupSeminormClass.toSeminormedCommGroup {F : Type u_1} {α : Type u_2} [FunLike F α ℝ] [CommGroup α] [GroupSeminormClass F α ℝ] (f : F) :
      SeminormedCommGroup α

      Constructs a SeminormedCommGroup structure from a GroupSeminormClass on a CommGroup.

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        @[reducible, inline]
        abbrev AddGroupSeminormClass.toSeminormedAddCommGroup {F : Type u_1} {α : Type u_2} [FunLike F α ℝ] [AddCommGroup α] [AddGroupSeminormClass F α ℝ] (f : F) :
        SeminormedAddCommGroup α

        Constructs a SeminormedAddCommGroup structure from an AddGroupSeminormClass on an AddCommGroup.

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          theorem GroupSeminormClass.toSeminormedCommGroup_norm_eq {F : Type u_1} {α : Type u_2} [FunLike F α ℝ] [CommGroup α] [GroupSeminormClass F α ℝ] (f : F) (x : α) :
          ‖x‖ = f x
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          theorem AddGroupSeminormClass.toSeminormedAddCommGroup_norm_eq {F : Type u_1} {α : Type u_2} [FunLike F α ℝ] [AddCommGroup α] [AddGroupSeminormClass F α ℝ] (f : F) (x : α) :
          ‖x‖ = f x
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          @[reducible, inline]
          abbrev GroupNormClass.toNormedGroup {F : Type u_1} {α : Type u_2} [FunLike F α ℝ] [Group α] [GroupNormClass F α ℝ] (f : F) :
          NormedGroup α

          Constructs a NormedGroup structure from a GroupNormClass on a Group.

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            @[reducible, inline]
            abbrev AddGroupNormClass.toNormedAddGroup {F : Type u_1} {α : Type u_2} [FunLike F α ℝ] [AddGroup α] [AddGroupNormClass F α ℝ] (f : F) :
            NormedAddGroup α

            Constructs a NormedAddGroup structure from an AddGroupNormClass on an AddGroup.

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              theorem GroupNormClass.toNormedGroup_norm_eq {F : Type u_1} {α : Type u_2} [FunLike F α ℝ] [Group α] [GroupNormClass F α ℝ] (f : F) (x : α) :
              ‖x‖ = f x
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              theorem AddGroupNormClass.toNormedAddGroup_norm_eq {F : Type u_1} {α : Type u_2} [FunLike F α ℝ] [AddGroup α] [AddGroupNormClass F α ℝ] (f : F) (x : α) :
              ‖x‖ = f x
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              @[reducible, inline]
              abbrev GroupNormClass.toNormedCommGroup {F : Type u_1} {α : Type u_2} [FunLike F α ℝ] [CommGroup α] [GroupNormClass F α ℝ] (f : F) :
              NormedCommGroup α

              Constructs a NormedCommGroup structure from a GroupNormClass on a CommGroup.

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                @[reducible, inline]
                abbrev AddGroupNormClass.toNormedAddCommGroup {F : Type u_1} {α : Type u_2} [FunLike F α ℝ] [AddCommGroup α] [AddGroupNormClass F α ℝ] (f : F) :
                NormedAddCommGroup α

                Constructs a NormedAddCommGroup structure from an AddGroupNormClass on an AddCommGroup.

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                  theorem GroupNormClass.toNormedCommGroup_norm_eq {F : Type u_1} {α : Type u_2} [FunLike F α ℝ] [CommGroup α] [GroupNormClass F α ℝ] (f : F) (x : α) :
                  ‖x‖ = f x
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                  theorem AddGroupNormClass.toNormedAddCommGroup_norm_eq {F : Type u_1} {α : Type u_2} [FunLike F α ℝ] [AddCommGroup α] [AddGroupNormClass F α ℝ] (f : F) (x : α) :
                  ‖x‖ = f x