complexOfReal_hasFDerivAt — Mathlib

English

The complexOfReal construction yields a complex-linear map whose action agrees with the original real-linear map.

Русский

Конструкция complexOfReal даёт комплексно-линейную карту, действующая так же, как исходная вещественно-линейная карта.

LaTeX

$$complexOfReal(\ell, h) is a map ℂ →L[ℂ] E extending \ell$$

Lean4

/-- In cases where the **Cauchy-Riemann Equation** guarantees complex differentiability at `x`, the
complex derivative equals `ContinuousLinearMap.complexOfReal` of the real derivative.
-/
protected theorem complexOfReal_hasFDerivAt {f' : ℂ →L[ℝ] E} (h₁ : HasFDerivAt f f' x) (h₂ : f' I = I • f' 1) :
    HasFDerivAt f (f'.complexOfReal h₂) x :=
  hasFDerivAt_of_restrictScalars ℝ h₁ rfl

Похожие утверждения