Theorems · Definition · functional analysis

NormedAddGroup.ofAddDist

{E : Type u_5} →
  [inst : Norm E] →
    [inst_1 : AddGroup E] →
      [inst_2 : MetricSpace E] →
        (∀ (x : E), ‖x‖ = dist 0 x) → (∀ (x y z : E), dist x y ≤ dist (z + x) (z + y)) → NormedAddGroup E

Construct a normed group from a translation-invariant distance.

Defined in: Mathlib.Analysis.Normed.Group.Defs
Cited by: 0 results in Mathlib
Foundations: Depth 103 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Norm AddGroup MetricSpace

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Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Realstatement · cited by 25,697
Norm.normstatement and proof · cited by 5,413
AddGroupstatement and proof · cited by 4,410
MetricSpacestatement and proof · cited by 1,684
Dist.diststatement and proof · cited by 1,539
Normstatement and proof · cited by 512
SeminormedAddGroupproof · cited by 331
NormedAddGroupstatement · cited by 32
SeminormedAddGroup.dist_eqproof · cited by 3
SeminormedAddGroup.ofAddDistproof · cited by 0

Cited by1

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NormedAddCommGroup.ofAddDistproof · cited by 0