apply_norm_eq_sqrt_inner_adjoint_right

English

Fundamental property: ⟪x, A† y⟫ = ⟪A x, y⟫ for every x ∈ E, y ∈ F.

Русский

Фундamental property: ⟪x, A† y⟫ = ⟪A x, y⟫, для всех x ∈ E, y ∈ F.

LaTeX

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

Lean4

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

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