ofBounds — Mathlib

English

Define a congruence-preserving map to compare LinearIsometryEquiv by their action on vectors.

Русский

Определим конгруэнтное соответствие между LinearIsometryEquiv по их действию на векторах.

LaTeX

$$Congruence-preserving map to compare LinearIsometryEquiv by action on vectors$$

Lean4

/-- Construct a `LinearIsometryEquiv` from a `LinearEquiv` and two inequalities:
`∀ x, ‖e x‖ ≤ ‖x‖` and `∀ y, ‖e.symm y‖ ≤ ‖y‖`. -/
def ofBounds (e : E ≃ₛₗ[σ₁₂] E₂) (h₁ : ∀ x, ‖e x‖ ≤ ‖x‖) (h₂ : ∀ y, ‖e.symm y‖ ≤ ‖y‖) : E ≃ₛₗᵢ[σ₁₂] E₂ :=
  ⟨e, fun x => le_antisymm (h₁ x) <| by simpa only [e.symm_apply_apply] using h₂ (e x)⟩

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