Finite-dimensionality transfers across a sphere isometry

MathlibAnnex.Sphere.finiteDimensional_codomain

theorem

Uses the radial homeomorphism to transfer local compactness, then applies the finite-dimensionality criterion for real normed spaces.

Statement

Let be real normed spaces. Put and , each with its ambient chord distance. Let be a bijective isometry of their unit spheres with the ambient chord metrics. If is finite-dimensional over , then is finite-dimensional over .

Assumptions

Only is assumed finite-dimensional at the start. Both spaces have real normed-vector-space structures, and is a sphere isometry equivalence. There is no initial completeness or finite-dimensionality assumption on ; the zero-dimensional case is included. The two carriers may have different universes in the formal statement.

Conclusion

The codomain is finite-dimensional. Applying the theorem to transfers finite-dimensionality in the opposite direction; that is the separately cited domain theorem.

Notes

The map used to carry the topology is the radial homeomorphism , with inverse . This step asserts transfer of finite-dimensionality; a linear isometry is not constructed here.

Proof route

For clarity, the radial map is and for , with inverse . Use the homeomorphism , whose continuity and inverse continuity follow from the two -Lipschitz estimates. Finite-dimensional real normed spaces are proper, so every closed bounded ball in is compact. Each point therefore has a compact neighbourhood, making locally compact. A homeomorphism carries a compact neighbourhood to a compact neighbourhood, so is locally compact. The local-compactness criterion for a real normed space now implies that is finite-dimensional:

theorem finiteDimensional_codomain
    [FiniteDimensional ℝ X]
    (e : Metric.sphere (0 : X) 1 ≃ᵢ Metric.sphere (0 : Y) 1) :
    FiniteDimensional ℝ Y

namespace MathlibAnnex
namespace Sphere

universe u v

variable {X : Type u} {Y : Type v}
  [NormedAddCommGroup X] [NormedSpace ℝ X]
  [NormedAddCommGroup Y] [NormedSpace ℝ Y]

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