apply_norm_eq_sqrt_inner_adjoint_left

English

The fundamental inner product identity for adjoint: ⟪A† y, x⟫ = ⟪y, A x⟫.

Русский

Фундamental inner product identity для сопряжённого: ⟪A† y, x⟫ = ⟪y, A x⟫.

LaTeX

$$$$ \\langle A^{\\dagger} y, x \\rangle = \\langle y, A x \\rangle. $$$$

Lean4

theorem apply_norm_eq_sqrt_inner_adjoint_left (A : E →L[𝕜] F) (x : E) : ‖A x‖ = √(re ⟪(A† ∘L A) x, x⟫) := by
  rw [← apply_norm_sq_eq_inner_adjoint_left, Real.sqrt_sq (norm_nonneg _)]

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