toContinuousLinearEquiv — Mathlib

English

The natural coe from a linear equivalence to a continuous linear equivalence is the identity on the map.

Русский

Естественное приведение линейного эквививалента к непрерывному линейному эквививаленту есть тождественный переход.

LaTeX

$$@[simp] (e : E ≃ₗ[𝕜] F) : (e.toContinuousLinearEquiv : E → F) = e$$

Lean4

/-- The continuous linear equivalence induced by a linear equivalence on a finite-dimensional
space. -/
def toContinuousLinearEquiv (e : E ≃ₗ[𝕜] F) : E ≃L[𝕜] F :=
  { e with
    continuous_toFun := e.toLinearMap.continuous_of_finiteDimensional
    continuous_invFun :=
      haveI : FiniteDimensional 𝕜 F := e.finiteDimensional
      e.symm.toLinearMap.continuous_of_finiteDimensional }

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