Proposition 17

Tag 0AFW, permanent. Cite it as [Bourbaki, Tag 0AFW].

Lie Groups and Lie Algebras, chapter III, § 6. Real and complex Lie groups

Let Δ be the set of z ∈ C such that −π < J(z) < π and Δ' the set of z ∈ C which are not real ≤ 0. Let E be a complete normable space over C and A (resp. A') the set of x ∈ L(E) whose spectrum Sp(x) is contained in Δ (resp. Δ'). Then A (resp. A') is an open subset of L(E) (resp. GL(E)) and the mappings exp: A → A' and log: A' → A (Spectral Theories, Chapter I, § 4, no. 9) are inverse isomorphisms of analytic manifolds.

This follows from Spectral Theories, Chapter I, § 4, Proposition 10 and no. 9.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Proposition 19, § 6
Theorem 6, § 6
Exercise 18, § 9
Proposition 17, § 9

Cites

Spectral Theories, Chapter I, § 4, Proposition 10 not yet on this site: TS I
Spectral Theories, Chapter I, § 4, no. 9 not yet on this site: TS I