By J Szczepaniak, H Blasinski, P Bogacz, B. Bogdanski

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**Example text**

37 that there exists an open neighborhood V of (0, 0, 0) ∈ g × g × g such that V ⊆ Ω and g1 |V = g2 |V . Since both g1 and g2 are analytic and Ω is connected, it then follows that g1 = g2 throughout on Ω. In fact, denote Ω0 = {ω ∈ Ω | (∀n ≥ 0) (n) (g1 )(n) ω = (g2 )ω }. Then Ω0 is clearly closed, and ∅ = V ⊆ Ω0 . 37. Since Ω is connected, we get Ω0 = Ω. In particular, g1 = g2 on Ω, as claimed. 34 Let g and h be (real or complex) contractive BanachLie algebras and ϕ : g → h a homomorphism of topological Lie algebras with ϕ ≤ 1.

H) = βn (h) for all h ∈ g × g. 33 algebra and denote Let g be a (real or complex) contractive Banach-Lie D := {(x, y) ∈ g × g | x + y < log 2}. Then H : D → g is a (real or complex) analytic mapping and H(x, H(y, z)) = H(H(x, y), z) whenever x + y + z < log(4/3). 29 and its proof. 34 (n) we easily deduce that H : D → g is smooth and H(0,0) = βn , whence (n) H(0,0) (h, . . 6) for all n ≥ 1 and h ∈ g × g. 37 that H is (real or complex) analytic. Next note that for x, y, z ∈ g with x + y + z < log(4/3) we have x + H(y, z) ≤ x − log 2 − e = log 2− e− x y + z e x ·e x + y + z x < log e 2 − (4/3)e− x < log 4/3 2 − (4/3)e− x < log 2, and similarly H(x, y) + z < log 2.

In particular, for t = 1 we get f2 (g) = f1 (g), as desired. 27 Let G be a Lie group and v : R → L(G) a smooth path. If there exists a left indefinite product integral p : R → G of v(·) (that is, δ l p = v), then the element Π(v) := p(0)−1 p(1) ∈ G is called the left definite product integral of v(·). 25, the element Π(v) ∈ G does not depend on the choice of the left indefinite product integral p(·) of v(·). 28 A Lie group G is said to be regular if it satisfies the following two conditions: (i) Every smooth path v : R → L(G) admits left indefinite product integrals.

### Discrete Mathematics Extensive Course - Prepared for IFE Students by J Szczepaniak, H Blasinski, P Bogacz, B. Bogdanski

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