toContinuousLinearMap₁ — Mathlib

English

Proofs show how bounds imply continuity for linear and semilinear maps.

Русский

Доказательства показывают, как границы влекут непрерывность линейных и семилинейных отображений.

LaTeX

$$$\text{Bounded implies continuous}$ for linear/semilinear maps$$

Lean4

/-- Reinterpret a linear map `𝕜 →ₗ[𝕜] E` as a continuous linear map. This construction
is generalized to the case of any finite-dimensional domain
in `LinearMap.toContinuousLinearMap`. -/
def toContinuousLinearMap₁ (f : 𝕜 →ₗ[𝕜] E) : 𝕜 →L[𝕜] E :=
  f.mkContinuous ‖f 1‖ fun x => by
    conv_lhs => rw [← mul_one x]
    rw [← smul_eq_mul, f.map_smul, mul_comm]; exact norm_smul_le _ _

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