instPseudoMetricSpace — Mathlib

English

An instance provides an emetric space on PiLp p α when each α_i is an emetric space, via the uniform structure coming from Pi.instT0Space.

Русский

Экземпляр задаёт эм метрическое пространство на PiLp p α, когда каждое α_i является эм метрическим пространством.

LaTeX

$$EMetricSpace (PiLp p α)$$

Lean4

/-- pseudometric space instance on the product of finitely many pseudometric spaces, using the
`L^p` distance, and having as uniformity the product uniformity. -/
instance [∀ i, PseudoMetricSpace (β i)] : PseudoMetricSpace (PiLp p β) :=
  ((pseudoMetricAux p β).replaceUniformity (uniformity_aux p β).symm).replaceBornology fun s =>
    Filter.ext_iff.1 (cobounded_aux p β).symm sᶜ

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