Local convergence of the Baker–Campbell–Hausdorff series

If $\mathfrak{g}=0$, every $H_N$ vanishes and the conclusion holds for every positive $\epsilon$. Otherwise choose one finite basis. By [F2]--[F3], its finitely many structure constants and the bounded coordinate maps give a constant $C\ge 0$ such that $\parallel [U,V]\parallel \le C\parallel U\parallel \parallel V\parallel$ for all $U,V\in \mathfrak{g}$. This chooses one finite witness, not a family.

Induction on word length now gives $\parallel [Z_1\cdots Z_N{]}_R\parallel \le C^{N-1}{\prod }_{j=1}^N\parallel Z_j\parallel$ for every right-nested commutator in [F1].

If $C=0$, all brackets vanish, so [F1] gives $H_1(X,Y)=X+Y$ and $H_N(X,Y)=0$ for $N\ge 2$; take $\epsilon =1$. Hence suppose $C>0$ and put $\epsilon =1\mathrm{/}(4C)$.

Fix $0<r_0<\epsilon$ and put $R=C{r}_0<1\mathrm{/}4$. For $a=C\parallel X\parallel$, $b=C\parallel Y\parallel$ and $(X,Y)\in D_{r_0}$, [F6] shows that the sum of the scalar weights $a^m{b}^n\mathrm{/}(m!n!)$ over a block of positive total degree is $q(a,b)=\exp (a+b)-1$, and its degree-$d$ part is $(a+b{)}^d\mathrm{/}d!\le R^d\mathrm{/}d!$. Thus these coefficients are bounded degree by degree by those of $q_0:={\sum }_{d\ge 1}{R}^d\mathrm{/}d!$. Since $R<1\mathrm{/}4$ and $d!\ge 1$, [F5]--[F7] give $0\le q_0\le {\sum }_{d\ge 1}(1\mathrm{/}4{)}^d=1\mathrm{/}3<1$.

Apply step 2.1 to the formula in [F1]. For a $k$-block summand of total degree $N$, absorb $C^N$ into its scalar letter weights and retain the factor $1\mathrm{/}C$. Since $1\mathrm{/}N\le 1$, the sum of the norms of all homogeneous terms, uniformly for $(X,Y)\in D_{r_0}$, is bounded by the nonnegative degree expansion of $\frac{1}{C}{\sum }_{k\ge 1}{q}_0^k\mathrm{/}k\le \frac{1}{C}{\sum }_{k\ge 1}{q}_0^k$, which converges by [F7]. In particular, the resulting degree majorants $M_N$ satisfy $\parallel H_N(X,Y)\parallel \le M_N$ on $D_{r_0}$ and ${\sum }_N{M}_N<\mathrm{\infty }$.

For each $(X,Y)\in D_{r_0}$, step 4.1 makes the partial sums Cauchy by the triangle inequality, and [F4] supplies their limit. Moreover, the norm of every tail is bounded by the corresponding scalar tail ${\sum }_{N>m}{M}_N$, independently of $(X,Y)$; that tail tends to zero. Hence the convergence is absolute and uniform on $D_{r_0}$.

Any pair with $\parallel X\parallel +\parallel Y\parallel <\epsilon$ lies in some $D_{r_0}$ with $\parallel X\parallel +\parallel Y\parallel <r_0<\epsilon$, so step 5.1 proves both claims. The zero and one-dimensional cases are included; in dimension one the bracket is zero. Degenerate brackets are allowed. There is no interval or endpoint, no metric beyond the arbitrary norm in the statement, no choice principle beyond choosing one finite basis, and no biconditional.