restrict_invariant — Mathlib

English

If T is symmetric and V is invariant under T, then the restriction T|V is symmetric.

Русский

Если T симметричен и у subмодуля V пределимый T, то ограничение T|V симметрично.

LaTeX

$$$ IsSymmetric(T|_V) $ when $T$ is symmetric and $V$ is invariant under $T$$$

Lean4

/-- If a symmetric operator preserves a submodule, its restriction to that submodule is
symmetric. -/
theorem restrict_invariant {T : E →ₗ[𝕜] E} (hT : IsSymmetric T) {V : Submodule 𝕜 E} (hV : ∀ v ∈ V, T v ∈ V) :
    IsSymmetric (T.restrict hV) := fun v w => hT v w

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