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THE aim of this chapter is to collect all the missing proofs of the results in 3 Chap. 2. The chapter is divided into several sections, corresponding to 4 those of Chap. 2. Finally, Sect. 7.8 collects some proofs from Chaps. 4 and 6 5 too, considered as less crucial in favor of economy of presentation in the 6 chapters they originally belonged to.

THE aim of this chapter is twofold. On the one hand, we aim to study the 4 convergence of the Dynkin series $$ \begin{array}{*{20}c} {uv: = } & {\sum\limits_{j = 1}^\infty {\left( {\sum\limits_{n = 1}^j {\frac{{\left( { - 1} \right)^{n + 1} }}{n}\,\sum\limits_{\begin{array}{*{20}c} {(h_1,k_1 ), \cdots (h_n,k_n ) \ne (0,0)} \\ {h_1 + k_1 + \cdots + h_n + k_n = j} \\ \end{array}} \times \frac{{(ad\,u)^{h_1 } (ad\,\upsilon )^{k_1 } \cdots (ad\,u)^{h_n } (ad\,\upsilon )^{k_n - 1} (\upsilon )}}{{h_1 ! \cdots h_n !k_1 ! \cdots k_n !(\sum\nolimits_{i = 1}^n {(h_i + k_i )} )}}} } \right),} } \\ \end{array} $$ in various contexts. For instance, this series can be investigated in any nilpotent Lie algebra (over a field of characteristic zero) where it is actually a finite sum, or in any finite dimensional real or complex Lie algebra and, more generally, its convergence can be studied in any normed Banach-Lie algebra (over R or C). For example, the case of the normed Banach algebras (becoming normed Banach-Lie algebras if equipped with the associated commutator) will be extensively considered here.

THE aim of this chapter is to give all the details of five other proofs (besides the one given in Chap. 3) of the Campbell, Baker, Hausdorff Theorem, stating that ; x♦y := Log(Exp(x) ·Exp(y)) is a series of Lie polynomials in x, y. As we showed in Chap. 3, this is the “qualitative” part of the CBHD Theorem, and the actual formula expressing xthat x♦y as an explicit series (that is, Dynkin’s Formula) can be quite easily derived from this qualitative counterpart as exhibited in Sect. 3.3.

THE aim of this chapter is to present the main proof of the Campbell- Baker-Hausdorff-Dynkin Theorem (CBHD for short), the topic of this Book. The proof is split into two very separate parts.

THE aim of this chapter is twofold. On the one hand (Sect. 8.1), we complete the missing proof from Chap. 2 concerning the existence of a free Lie algebra Lie(X) related to a set X. This proof relies on the direct construction of Lie(X) as a quotient of the free non-associative algebra Lib(X). Furthermore, we prove that Lie(X) is isomorphic to L(K_X_), and the latter provides a free Lie algebra over X.

THE aim of this chapter is to collect some prerequisites on formal power series in one indeterminate, needed in this Book. One of the main aims is to furnish a purely algebraic proof of the fact that, by substituting into each other – in any order – the two series $$\sum^\infty_{n=1} \frac{x^n}{n!} \quad {\rm and} \quad \sum^\infty_{n=1} \frac{(-1)^{(n+1)}x^n}{n}$$ one obtains the result x.

THE aim of this chapter is to unravel the close relationship existing = between the Theorems of CBHD and of Poincare-Birkhoff-Witt (“PBW” for short) and to show how the existence of free Lie algebras intervenes.We have analyzed, in Chap. 3, how the PBWTheoremintervenes in the classical approach to the proof of CBHD, in that PBWcan be used to prove in a simple way Friedrichs’s characterization of L(V ) (see Theorem 3.13 on page 133). Also, as for the proofs of CBHD in Chap. 4, the rOle of the free Lie algebras was broadly manifest.