comp — Mathlib

English

If f ∈ C0(γ, δ) and g is a cocompact map β → γ, then the composed map f ∘ g lies in C0(β, δ).

Русский

Если f ∈ C0(γ, δ) и g — кокомпактная отображение β → γ, то композиция f ∘ g принадлежит C0(β, δ).

LaTeX

$$$f \\in C_0(\\gamma, \\delta) \\ \quad g : \\beta \\to_{\\text{co}} \\gamma \\quad \\Rightarrow \\quad f \\circ g \\in C_0(\\beta, \\delta)$$$

Lean4

/-- Composition of a continuous function vanishing at infinity with a cocompact map yields another
continuous function vanishing at infinity. -/
def comp (f : C₀(γ, δ)) (g : β →co γ) : C₀(β, δ)
    where
  toContinuousMap := (f : C(γ, δ)).comp g
  zero_at_infty' := (zero_at_infty f).comp (cocompact_tendsto g)

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