adjoint_inner_left — Mathlib

English

For a star-normal operator T, the equation adjoint T x = 0 is equivalent to T x = 0 for all x.

Русский

Для нормального по звёздочке оператора T равенство adjoint T x = 0 эквивалентно T x = 0 для любого x.

LaTeX

$$$\\text{adjoint } T \\, x = 0 \\iff T x = 0$$$

Lean4

/-- The fundamental property of the adjoint. -/
theorem adjoint_inner_left (A : E →ₗ[𝕜] F) (x : E) (y : F) : ⟪adjoint A y, x⟫ = ⟪y, A x⟫ :=
  by
  have := FiniteDimensional.complete 𝕜 E
  have := FiniteDimensional.complete 𝕜 F
  rw [← coe_toContinuousLinearMap A, adjoint_eq_toCLM_adjoint]
  exact ContinuousLinearMap.adjoint_inner_left _ x y

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