Theorems · Theorem · functional analysis
spectrum.norm_le_norm_mul_of_mem
∀ {𝕜 : Type u_1} {A : Type u_2} [inst : NormedField 𝕜] [inst_1 : NormedRing A] [inst_2 : NormedAlgebra 𝕜 A]
[CompleteSpace A] {a : A} {k : 𝕜}, k ∈ spectrum 𝕜 a → ‖k‖ ≤ ‖a‖ * ‖1‖- Defined in
- Mathlib.Analysis.Normed.Algebra.Spectrum
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 172 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
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
- spectrum.mem_resolventSet_of_norm_lt_mulproof · cited by 3
Cited by4
Results whose statement or proof uses this declaration.
- spectrum.subset_closedBall_norm_mulproof · cited by 2
- spectrum.spectralRadius_le_pow_nnnorm_pow_one_divproof · cited by 1
- AlgHom.norm_apply_le_self_mul_norm_oneproof · cited by 0
- WeakDual.CharacterSpace.norm_le_norm_oneproof · cited by 0