Theorems · Theorem · functional analysis
IsStrictlyPositive.spectrum_pos
∀ {A : Type u_1} {𝕜 : Type u_2} [inst : Ring A] [inst_1 : PartialOrder A] [inst_2 : CommSemiring 𝕜]
[inst_3 : PartialOrder 𝕜] [inst_4 : Algebra 𝕜 A] [NonnegSpectrumClass 𝕜 A] {a : A},
IsStrictlyPositive a → ∀ {x : 𝕜}, x ∈ spectrum 𝕜 a → 0 < x- Defined in
- Mathlib.Algebra.Algebra.StrictPositivity
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 44 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- Ringstatement and proof · cited by 7,463
- PartialOrderstatement and proof · cited by 6,410
- spectrumstatement and proof · cited by 510
- NonnegSpectrumClassstatement and proof · cited by 292
- lt_of_le_of_neproof · cited by 230
- IsStrictlyPositivestatement and proof · cited by 75
Cited by4
Results whose statement or proof uses this declaration.
- CStarAlgebra.isUnit_of_leproof · cited by 3
- cfcHom_isStrictlyPositive_iffproof · cited by 1
- CStarAlgebra.convexOn_ringInverseproof · cited by 1
- CFC.tendsto_cfc_rpow_sub_one_logproof · cited by 1