instPseudoMetricSpace — Mathlib

English

The space of bounded continuous functions, with the uniform distance, forms a pseudometric space.

Русский

Пространство ограниченно непрерывных функций с единичным расстоянием образует псевдометрическое пространство.

LaTeX

$$The type of bounded continuous functions with the uniform distance is a pseudometric space.$$

Lean4

/-- The type of bounded continuous functions, with the uniform distance, is a pseudometric space. -/
instance instPseudoMetricSpace : PseudoMetricSpace (α →ᵇ β)
    where
  dist_self f := le_antisymm ((dist_le le_rfl).2 fun x => by simp) dist_nonneg'
  dist_comm f g := by simp [dist_eq, dist_comm]
  dist_triangle _ _
    _ :=
    (dist_le (add_nonneg dist_nonneg' dist_nonneg')).2 fun _ =>
      le_trans (dist_triangle _ _ _) (add_le_add (dist_coe_le_dist _) (dist_coe_le_dist _))

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