Theorems · Theorem · functional analysis
NormedSpace.exp_add_of_commute
∀ {𝔸 : Type u_1} [inst : NormedRing 𝔸] [NormedAlgebra ℚ 𝔸] [CompleteSpace 𝔸] {x y : 𝔸},
Commute x y → NormedSpace.exp (x + y) = NormedSpace.exp x * NormedSpace.exp yIn a Banach-algebra 𝔸 over 𝕂 = ℝ or 𝕂 = ℂ, if x and y commute, then
NormedSpace.exp (x+y) = (NormedSpace.exp x) * (NormedSpace.exp y).
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 177 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CompleteSpacestatement and proof · cited by 2,532
- NormedAlgebrastatement and proof · cited by 1,165
- NormedRingstatement and proof · cited by 924
- Commutestatement and proof · cited by 639
- NormedSpace.expstatement · cited by 157
- edist_lt_topproof · cited by 32
- NormedSpace.expSeries_radius_eq_topproof · cited by 27
- NormedSpace.exp_add_of_commute_of_mem_ballproof · cited by 3
Cited by8
Results whose statement or proof uses this declaration.
- Quaternion.exp_eqproof · cited by 3
- NormedSpace.exp_sum_of_commuteproof · cited by 2
- NormedSpace.exp_nsmulproof · cited by 2
- spectrum.exp_mem_expproof · cited by 1
- Commute.expUnitary_addproof · cited by 1
- NormedSpace.exp_mem_unitary_of_mem_skewAdjointproof · cited by 1
- isStarNormal_iff_forall_exp_mul_exp_mem_unitaryproof · cited by 0
- Matrix.exp_add_of_commuteproof · cited by 0