English
Let ha be a QuasispectrumRestricts a ContinuousMap.realToNNReal. Then for all r ∈ ℝ≥0, the inequality with ≤ holds equivalently for quasispectrum in NNReal and spectrum in Real.
Русский
Пусть ha — QuasispectrumRestricts a ContinuousMap.realToNNReal. Тогда для всех r ∈ ℝ≥0 неравенство ≤ между квази-спектром NNReal и спектром Real эквивалентно друг другу.
LaTeX
$$$\forall r \in \mathbb{R}_{\ge 0},\; (\forall x \in quasispectrum_{\mathbb{R}_{\ge 0}} a, x \le r) \iff (\forall x \in quasispectrum_{\mathbb{R}} a, x \le r)$$$
Lean4
/-- Alternative constructor for a `Seminorm` on an `AddCommGroup E` that is a module over a
`SeminormedRing 𝕜`. -/
def of [SeminormedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] (f : E → ℝ) (add_le : ∀ x y : E, f (x + y) ≤ f x + f y)
(smul : ∀ (a : 𝕜) (x : E), f (a • x) = ‖a‖ * f x) : Seminorm 𝕜 E
where
toFun := f
map_zero' := by rw [← zero_smul 𝕜 (0 : E), smul, norm_zero, zero_mul]
add_le' := add_le
smul' := smul
neg' x := by rw [← neg_one_smul 𝕜, smul, norm_neg, ← smul, one_smul]
