Theorems · Definition · general topology

ContinuousMap.isometryEquivBoundedOfCompact

(α : Type u_1) →
  (β : Type u_2) →
    [inst : TopologicalSpace α] →
      [inst_1 : CompactSpace α] → [inst_2 : PseudoMetricSpace β] → C(α, β) ≃ᵢ BoundedContinuousFunction α β

When α is compact, and β is a metric space, the bounded continuous maps α →ᵇ β are isometric to C(α, β).

Defined in

Mathlib.Topology.ContinuousMap.Compact

Cited by

8 results in Mathlib

Foundations

Depth 149 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceCompactSpacePseudoMetricSpace