English
Let ha be a QuasispectrumRestricts a ContinuousMap.realToNNReal. Then for any r ∈ ℝ≥0, the following equivalence holds between NNReal- and Real-spectrum inequalities: (∀ x ∈ spectrum NNReal a, x ≤ r) ↔ (∀ x ∈ spectrum Real a, x ≤ r).
Русский
Пусть ha — QuasispectrumRestricts a ContinuousMap.realToNNReal. Тогда ∀ r ∈ ℝ≥0 верна эквивалентность между неравенствами спектра NNReal и спектра Real: (∀ x ∈ spectrum NNReal a, x ≤ r) ↔ (∀ x ∈ spectrum Real a, x ≤ r).
LaTeX
$$$\forall r \in \mathbb{R}_{\ge 0},\; (\forall x \in spectrum_{\mathbb{R}_{\ge 0}} a, x \le r) \iff (\forall x \in spectrum_{\mathbb{R}} a, x \le r)$$$
Lean4
instance instFunLike : FunLike (Seminorm 𝕜 E) E ℝ
where
coe f := f.toFun
coe_injective' f g
h := by
rcases f with ⟨⟨_⟩⟩
rcases g with ⟨⟨_⟩⟩
congr
