Theorems · Theorem · functional analysis
spectrum.norm_le_norm_of_mem
∀ {𝕜 : Type u_1} {A : Type u_2} [inst : NormedField 𝕜] [inst_1 : NormedRing A] [inst_2 : NormedAlgebra 𝕜 A]
[CompleteSpace A] [NormOneClass A] {a : A} {k : 𝕜}, k ∈ spectrum 𝕜 a → ‖k‖ ≤ ‖a‖- Defined in
- Mathlib.Analysis.Normed.Algebra.Spectrum
- Cited by
- 11 results in Mathlib
- Foundations
- Depth 173 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Realstatement · cited by 25,697
- Norm.normstatement · cited by 5,413
- CompleteSpacestatement and proof · cited by 2,532
- NormedAlgebrastatement and proof · cited by 1,165
- NormedFieldstatement and proof · cited by 1,084
- NormedRingstatement and proof · cited by 924
- spectrumstatement and proof · cited by 510
- le_of_not_gtproof · cited by 430
- NormOneClassstatement and proof · cited by 136
- spectrum.mem_resolventSet_of_norm_ltproof · cited by 1
Cited by11
Results whose statement or proof uses this declaration.
- IsSelfAdjoint.le_algebraMap_norm_selfproof · cited by 7
- CStarAlgebra.norm_le_iff_le_algebraMapproof · cited by 5
- selfAdjoint.norm_sq_expUnitary_sub_oneproof · cited by 3
- Unitary.spectrum_subset_circleproof · cited by 2
- spectrum.le_nnnorm_of_memproof · cited by 2
- spectrum.spectralRadius_le_nnnormproof · cited by 1
- AlgHom.norm_apply_le_selfproof · cited by 0
- argSelfAdjoint_expUnitaryproof · cited by 0
- AlgHom.toContinuousLinearMap_normproof · cited by 0
- spectrum.subset_closedBall_normproof · cited by 0
- IsSelfAdjoint.self_add_I_smul_cfcSqrt_sub_sq_mem_unitaryproof · cited by 0