Hilbert's Fifth Problem and Infinite-Dimensional Lie Theory

Introduction

The Lie correspondence of the companion article The Lie Correspondence and the Adjoint Representation is a theorem about smooth groups: the group operations are assumed smooth, the exponential is assumed to be the flow of a smooth vector field, and the algebra is recovered from the smooth structure. Hilbert's fifth problem asks whether the smoothness is needed at all: is every topological group that is locally Euclidean — so that it looks like $\mathbb{R}^n$ near the identity, but its group operations are only continuous — necessarily a Lie group? The answer is yes, and the theorem is a genuine one: continuity together with local compactness forces a smooth structure, and the smooth structure is unique. The proof required the notion of a group with no small subgroups and the theorems of Gleason and Montgomery–Zippin, supplemented by Yamabe's structure theorem for locally compact groups.

The same circle of ideas opens the infinite-dimensional theory, where the classical correspondence fails. A Banach–Lie group is still locally like a Banach space and its exponential map is still a local diffeomorphism, so the finite-dimensional theory survives. But for Fréchet–Lie groups, and in particular for the diffeomorphism group of a manifold, the exponential map is not a local diffeomorphism, the Campbell–Baker–Hausdorff series need not converge, and the algebra does not determine the group. The algebraic content of infinite-dimensional Lie theory is the study of these failures and of the additional hypotheses — regularity, tameness — under which the classical theorems can be recovered.

This article states Hilbert's fifth problem and its solution, develops the no-small-subgroups condition, states the structure theorem for locally compact groups, and then turns to the infinite-dimensional theory: Banach–Lie groups, the failure of the local diffeomorphism property for Fréchet–Lie groups, the diffeomorphism group with its Lie algebra of vector fields, and the Nash–Moser framework. Throughout, topological groups are Hausdorff topological groups with continuous multiplication and inversion, locally compact groups are Hausdorff, Lie groups are over $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$ in the sense of Lie Groups, and Lie algebras are written in lowercase fraktur, so $\mathrm{G} = T_eG$ and $\mathrm{X}(M)$ is the Lie algebra of vector fields on a manifold $M$. No physics is invoked.

Hilbert's Fifth Problem

Statement

Hilbert's fifth problem, posed in 1900, asks:

Is every topological group whose underlying topological space is a topological manifold necessarily a Lie group?

In the form in which it was posed, the group operation is assumed only continuous, and the question is whether a smooth structure compatible with the group operations exists and is unique. The affirmative answer rests on three separate facts: that a locally Euclidean topological group has no small subgroups, that a locally compact group with no small subgroups is a Lie group, and that a locally compact group is topologically a projective limit of Lie groups.

Definition. A topological group is a group $G$ with a topology for which the multiplication $G \times G \to G$ and the inversion $G \to G$ are continuous. A Lie group is a group that is a smooth manifold for which the same two maps are smooth.

Definition. A topological group $G$ has no small subgroups, abbreviated NSS, if there is a neighbourhood $U$ of the identity containing no subgroup of $G$ other than $\{e\}$; equivalently, every neighbourhood of the identity contains a neighbourhood with the same property. The condition is a local one, and it is inherited by open subgroups.

Example. A Lie group has NSS. In a neighbourhood of the identity small enough that the exponential is a local diffeomorphism, a subgroup is a union of curves through the identity tangent to its Lie algebra, and a nontrivial such subgroup has a nonzero tangent vector whose powers leave every sufficiently small neighbourhood. The precise statement is the one recorded below as a lemma.

The Role of Differentiability

The problem separates into a topological part and a smooth part.

Lemma. Let $G$ be a locally Euclidean topological group. Then $G$ is locally compact and has no small subgroups.

Proof sketch. Local compactness is immediate. For NSS, suppose on the contrary that there are nontrivial subgroups $H_k$ contained in balls of radius $1/k$ around $e$. Choosing $x_k \in H_k \setminus \{e\}$ with $x_k \to e$, one uses the local Euclidean structure to choose the $x_k$ in a coordinate ball and to compare the closure of the subgroup generated by $x_k$ with the number of its elements; the geometry of a small ball in $\mathbb{R}^n$ bounds the size of a finite subgroup acting by left translations, and the bound contradicts the existence of pairwise distinct powers of arbitrarily small elements.

The lemma is the reduction of Hilbert's problem to a purely infinite-group-theoretic statement: it suffices to know that a locally compact NSS group is a Lie group. This is the form in which the problem was solved.

The No-Small-Subgroups Condition

Proposition. The class of NSS groups is closed under taking open subgroups, taking closed subgroups, taking quotients by closed normal subgroups, and taking finite direct products.

Proof. An open subgroup inherits the neighbourhood. A closed subgroup $H$ does too, since a subgroup of $H$ lying in the subgroup-free neighbourhood $U \cap H$ is a subgroup of $G$ lying in $U$, hence trivial. A quotient map carries a subgroup-free neighbourhood to a subgroup-free neighbourhood, because the preimage of a subgroup is a subgroup. A product is handled by a product of subgroup-free neighbourhoods of the identities: the projection of a subgroup of the product into each factor is a subgroup of that factor, hence trivial.

The Solution of the Problem

The Theorems of Gleason and Montgomery–Zippin

Theorem (Gleason; Montgomery–Zippin). Let $G$ be a locally compact topological group with no small subgroups. Then $G$ is a Lie group: its topology is that of a smooth manifold, and the manifold structure is unique, so that $G$ is a Lie group in the sense of Lie Groups.

The theorem was proved independently by Gleason and by Montgomery and Zippin in 1952. Its proof is deep and proceeds by constructing a local coordinate system from the group operations and a one-parameter subgroup, and then verifying the smoothness of the group law in those coordinates.

Corollary (Hilbert's fifth problem, locally Euclidean case). Every topological group whose underlying space is a topological manifold is a Lie group, and its smooth structure is unique.

Proof. By the lemma, a locally Euclidean topological group is locally compact and has no small subgroups; the theorem of Gleason and Montgomery–Zippin applies.

Corollary. Every compact topological group is a Lie group if and only if it has no small subgroups; a compact group that is also locally Euclidean is a Lie group.

The Strategy of the Solution

The proof of the Gleason–Montgomery–Zippin theorem is organised in three steps, and the description is worth recording because it explains which hypotheses are used.

Step 1: one-parameter subgroups. Compactness and local compactness supply, near the identity, enough one-parameter subgroups: the closed subgroups generated by a single small element are homeomorphic to closed intervals when the element is small enough, and their union fills a neighbourhood of the identity. This uses the no-small-subgroups hypothesis to rule out the accumulation of torsion elements.

Step 2: a local coordinate system. A maximal family of independent one-parameter subgroups is chosen, and the multiplication map from a product of intervals in these subgroups to the group is shown to be a homeomorphism onto a neighbourhood of the identity. The dimension is the cardinality of the family and is finite because the group is locally compact and NSS.

Step 3: smoothness. In these coordinates the group law is continuous and satisfies the associativity identities; Gleason's lemma on the differentiability of continuous homomorphisms of one-parameter subgroups into a locally compact NSS group, together with the identities, upgrades continuity to smoothness and gives the Lie group structure. Uniqueness of the smooth structure follows because two smooth structures with the same one-parameter subgroups have the same exponential map.

Remark. Only Step 3 uses NSS; the first two steps use local compactness alone. This is why the general locally compact group can still be described as a projective limit of Lie groups near the identity.

Compact Groups and Inverse Limits

Theorem (structure of compact groups). Every compact topological group $G$ is a projective limit of compact Lie groups: there is an inverse system of compact Lie groups $G_i$ with continuous surjections $G \to G_i$ whose kernels are closed normal subgroups $N_i \subseteq G$ with $G/N_i$ a Lie group, and the natural map $G \to \varprojlim G_i$ is an isomorphism of topological groups.

Proof sketch. The Peter–Weyl theorem decomposes the regular representation of $G$ into finite-dimensional irreducible representations, and the kernel of each finite-dimensional representation is a closed normal subgroup with Lie quotient. The quotients by these kernels are compact Lie groups, the system of quotients is directed, and their inverse limit receives a continuous bijection from $G$; compactness makes the bijection a homeomorphism.

Corollary. A compact topological group is a Lie group exactly when it has no small subgroups, and exactly when the inverse system of compact Lie group quotients is eventually constant.

Totally Disconnected Groups and van Dantzig's Theorem

Definition. A topological group is totally disconnected if the only connected subset containing the identity is $\{e\}$; it is almost connected if the quotient by its identity component is compact.

Theorem (van Dantzig). Every totally disconnected locally compact group has a compact open subgroup.

Example. The additive group $\mathbb{Z}_p$ of $p$-adic integers, with the topology generated by the subgroups $p^k\mathbb{Z}_p$, is compact, totally disconnected and abelian; it is the projective limit of the finite groups $\mathbb{Z}/p^k\mathbb{Z}$, and every neighbourhood of the identity contains one of the subgroups $p^k\mathbb{Z}_p$. It is therefore locally compact but not a Lie group, and it lies outside Hilbert's fifth problem because it is not locally Euclidean.

Example. The 2-adic solenoid $S = \varprojlim(\mathbb{R}/\mathbb{Z} \xrightarrow{\times 2} \mathbb{R}/\mathbb{Z})$ is compact and connected, and it is a projective limit of copies of the circle, so it is locally a projective limit of Lie groups; it is not a Lie group, and hence, by the no-small-subgroups theorem, it has small subgroups. It shows that a compact connected group that is a projective limit of Lie groups need not be a Lie group, so the NSS hypothesis in the Gleason–Montgomery–Zippin theorem cannot be omitted.

The Hilbert–Smith Conjecture

Conjecture (Hilbert–Smith). If a locally compact topological group acts faithfully and continuously on a connected topological manifold of positive dimension, then the group is a Lie group.

Remark. The conjecture is the transformation-group form of the fifth problem: it asks not that the group itself be a manifold, but that an action on a manifold forces the group to be smooth. It is known when the manifold has dimension $1$ or $2$ by the classical theory of transformation groups, and in dimension $3$ by work of Pardon. The case of general dimension is open. For a group acting on itself by left translations the conjecture reduces to Hilbert's fifth problem, which is why the two questions are frequently discussed together.

Yamabe's Structure Theorem

Gleason and Montgomery–Zippin's theorem applies to NSS groups, and the general locally compact group is not NSS. The general structure is described by Yamabe's theorem, which says that the obstruction is compact.

Theorem (Yamabe). Let $G$ be a locally compact topological group. Then $G$ has an open subgroup $G_0$ that is a projective limit of Lie groups; equivalently, there is a compact normal subgroup $N$ of $G_0$ such that $G_0/N$ is a Lie group. If $G$ is almost connected (the quotient by the identity component is compact), then $G$ itself is a projective limit of Lie groups.

Corollary. Every locally compact topological group is, near the identity, a projective limit of Lie groups, and every compact topological group is a projective limit of compact Lie groups.

Remark. The corollary explains the boundary of Hilbert's problem. Compact groups that are not Lie groups, such as a countably infinite product of copies of a finite group with the product topology, are locally compact but not locally Euclidean; they are projective limits of finite groups rather than Lie groups, and they are not manifolds. The locally Euclidean hypothesis excludes them; locally Euclidean groups are NSS, and NSS groups are Lie groups.

Infinite-Dimensional Lie Theory

The Failure of the Classical Correspondence

The theorems above depend on local compactness, and they fail for infinite-dimensional groups. The manifold model is replaced by a topological vector space, and the group operations are required to be smooth in that model.

Definition. Let $E$ be a topological vector space and $G$ a group that is a manifold modelled on $E$ for which the multiplication and inversion are smooth. Then $G$ is a topological-vector-space Lie group, of Banach type if $E$ is a Banach space, of Fréchet type if $E$ is a Fréchet space, and of locally convex type if $E$ is a general locally convex space. The Lie algebra of $G$ is $\mathrm{G} = T_eG$ with the bracket obtained as in the finite-dimensional case from the left-invariant vector fields.

Remark. The definition of the Lie algebra is the same as in The Lie Algebra and the Exponential Map, and the functoriality of the construction is unchanged. What changes is the analysis: the inverse function theorem, the existence of flows, and the convergence of the Baker–Campbell–Hausdorff series are all statements about the model space, and they have different answers for Banach and for Fréchet spaces.

Banach–Lie Groups

Theorem. Let $G$ be a Banach–Lie group with Lie algebra $\mathrm{G}$, and let $X \in \mathrm{G}$. Then there is a unique one-parameter subgroup $\gamma_X : \mathbb{R} \to G$ with $\gamma_X'(0) = X$, obtained as the flow of the left-invariant field extending $X$; the exponential map $\exp(X) = \gamma_X(1)$ is defined, smooth, and a local diffeomorphism at $0$.

Proof. A smooth vector field on a Banach manifold has a local flow, by the Banach-space version of the existence and uniqueness theorem for ordinary differential equations; left invariance makes the flow global, since the flow at time $s$ of the flow at time $t$ is defined and equals the flow at time $s+t$. The differential at the origin is the identity, and the inverse function theorem for Banach spaces gives the local diffeomorphism.

Theorem. The Campbell–Baker–Hausdorff series converges in a neighbourhood of $0$ in $\mathrm{G} \times \mathrm{G}$ for a Banach–Lie algebra, and the Lie correspondence holds: the simply connected Banach–Lie groups form a category equivalent to that of Banach–Lie algebras.

Proof sketch. The convergence of the series uses the absolute convergence of the BCH series for elements of norm below a bound depending on the norm of the bracket, which is finite for a Banach–Lie algebra; the equivalence is then obtained exactly as in the finite-dimensional case.

Remark. For Banach–Lie groups the finite-dimensional theory therefore survives intact: exponential coordinates exist, the group law is a Lie polynomial, subalgebras integrate to subgroups, and the simply connected group attached to the algebra is unique. The hypotheses that make this work are completeness and a single norm, which supply the inverse function theorem and the convergence of the exponential series.

Fréchet and Beyond: Milnor's Theory

Theorem (failure of the local diffeomorphism property). There exist Fréchet–Lie groups $G$ and points $X \in \mathrm{G}$ at which the exponential map is not a local diffeomorphism, and there exist Fréchet–Lie groups whose exponential map is injective but not surjective onto any neighbourhood of the identity. The Campbell–Baker–Hausdorff series need not converge.

Example. Let $M$ be a compact smooth manifold and let $G = \operatorname{Diff}(M)$ be its diffeomorphism group, with the $C^\infty$ topology. Then $G$ is a Fréchet–Lie group, modelled on the space $\mathrm{X}(M)$ of smooth vector fields, and the exponential map is the time-one map of the flow of a vector field. It is not a local diffeomorphism at every point: at a nonzero vector field whose flow has period $1$ the derivative $d(\exp)_X$ is singular. Moreover the exponential is not locally surjective, its image containing no neighbourhood of the identity. These are genuine phenomena, not artefacts of the topology, and they show that the finite-dimensional theory does not extend to the diffeomorphism group.

Definition (Milnor). A (locally convex) Lie group $G$ is regular if the following hold: for every smooth curve $x : [0,1] \to \mathrm{G}$ there is a smooth curve $g : [0,1] \to G$ with $g(0) = e$ and the right logarithmic derivative $g'(t)g(t)^{-1} = x(t)$, and the correspondence is smooth. Equivalently, the BCH product is defined and smooth on a neighbourhood of $0$ in $\mathrm{G} \times \mathrm{G}$.

Theorem (Milnor). For a regular Lie group the group structure is determined by the Lie algebra: the BCH product is a smooth function of the algebra variables and gives the germ of the group law in exponential coordinates. The group $\operatorname{Diff}(M)$ of a compact manifold is regular, as is the group of smooth maps from a compact manifold to a finite-dimensional Lie group.

Proof sketch. The existence and smoothness of the solution of the differential equation in the definition is established by the theory of the Maurer–Cartan equation; the smoothness of the BCH product follows from the smoothness of the product in the group via the exponential, and the compactness of $M$ supplies the completeness needed for the flow of a smooth time-dependent vector field.

Remark. Not every Fréchet–Lie group is regular; there are examples in which the differential equation of the definition has no solution for some smooth curve $x$. Regularity is therefore a genuine extra hypothesis, and one of the main technical themes of infinite-dimensional Lie theory is to identify the groups for which it holds.

The BCH Series and Its Convergence

Definition. The Campbell–Baker–Hausdorff series of elements $X, Y$ of a Lie algebra is the formal series $Z(X,Y)$ in the free Lie algebra on two generators characterised by the identity

$$ \exp(X)\exp(Y) = \exp\bigl(Z(X,Y)\bigr), $$

whose first terms are

$$ Z(X,Y) = X + Y + \tfrac{1}{2}[X,Y] + \tfrac{1}{12}\bigl[X,[X,Y]\bigr] - \tfrac{1}{12}\bigl[Y,[Y,X]\bigr] + \cdots . $$

Theorem. For a finite-dimensional or Banach Lie algebra the series converges for $X, Y$ in a neighbourhood of the origin whose size is governed by the norm of the bracket, and its sum is analytic in the two variables; the products of exponentials are then given by a convergent Lie polynomial. For a Fréchet–Lie algebra the series can fail to converge in every neighbourhood of the origin.

Remark. The convergence of the BCH series is the property that makes the finite-dimensional and Banach theories algebraic: it is what allows the germ of the group law to be read off from the algebra. Its failure in the Fréchet setting is the reason the notion of a regular Lie group is formulated in terms of a differential equation rather than in terms of the convergence of a series.

Lie Algebras Without Groups

Remark. Not every infinite-dimensional Lie algebra is the Lie algebra of a corresponding group. The BCH series produces only a local group, and there are Banach–Lie algebras that enlarge to no global Banach–Lie group; the equivalence of the finite-dimensional Lie correspondence therefore has no unrestricted infinite-dimensional counterpart. The hypotheses that restore the correspondence are completeness of the model space together with regularity of the group, and even then the functor from groups to algebras is not an equivalence: non-isomorphic Lie groups can share a Lie algebra, exactly as in the finite-dimensional case, but the supply of quotients and subgroups that produced the classification is no longer controlled by the algebra alone.

The Exponential Map in Infinite Dimensions

The Local Diffeomorphism Property

The differential of the exponential is the operator

$$ d(\exp)_X = \sum_{k \geq 0} \frac{(-1)^k}{(k+1)!}\, (\operatorname{ad}_X)^k $$

formally, as in The Lie Algebra and the Exponential Map. The formula is meaningful whenever the operator series converges; for a Banach–Lie algebra it converges for every $X$, and for a Fréchet–Lie algebra it need not converge.

Theorem. For a Banach–Lie group, $\exp$ is a local diffeomorphism at $0$, and hence at every $X$ where $d(\exp)_X$ is invertible. For a Fréchet–Lie group both statements can fail.

The failure is the central obstruction to the classical correspondence: without the local diffeomorphism property there are no exponential coordinates, and the group law cannot be recovered from the algebra by a convergent Lie polynomial.

The Nash–Moser Setting

Definition. A tame Fréchet space is a Fréchet space with a family of seminorms satisfying tame estimates, a quantified version of the loss of derivatives in the composition of nonlinear maps. A tame map between tame Fréchet spaces is a smooth map satisfying tame estimates for itself and all its derivatives.

Theorem (Nash–Moser inverse function theorem). Let $F : E \to F$ be a tame smooth map between tame Fréchet spaces whose derivative $dF(x)$ is invertible for all $x$ in a neighbourhood of a point $x_0$, with the inverse satisfying tame estimates. Then $F$ is locally invertible near $x_0$.

Remark. The theorem is the infinite-dimensional replacement for the inverse function theorem in the Fréchet setting, and it is the technical device by which regularity statements for diffeomorphism groups are proved. It does not produce a local diffeomorphism for the exponential map of $\operatorname{Diff}(M)$ — that map is genuinely not a local diffeomorphism — but it produces the smoothness of the flow map and of the solutions of the Maurer–Cartan equation, which is what the definition of regularity requires.

Groups of Diffeomorphisms

The Diffeomorphism Group

Theorem. Let $M$ be a compact smooth manifold. Then the group $\operatorname{Diff}(M)$ of smooth diffeomorphisms of $M$, with the topology of uniform convergence of all derivatives, is a Fréchet–Lie group modelled on the Fréchet space $\mathrm{X}(M)$ of smooth vector fields. Its Lie algebra has bracket the negative of the bracket of vector fields, or the bracket of vector fields according to the convention for the right or left trivialisation.

Proof sketch. The space $C^\infty(M, M)$ of smooth maps is a Fréchet manifold, and $\operatorname{Diff}(M)$ is an open subset of it because the invertible maps form an open set for the $C^1$ topology; the group operations are smooth because composition of smooth maps is smooth as a map of Fréchet spaces. The tangent space at the identity is the space of smooth vector fields, identified with the derivations of $C^\infty(M)$.

Definition. The exponential map of $\operatorname{Diff}(M)$ sends a vector field $X$ to the time-one map of its flow, $\exp(X) = \Phi_X^1$. It is defined for all $X$ when $M$ is compact, because the flow of a smooth vector field on a compact manifold is complete.

The Lie Algebra of Vector Fields

Theorem. The space $\mathrm{X}(M)$ of smooth vector fields on $M$, with the bracket $[X, Y] = X \circ Y - Y \circ X$ of derivations of $C^\infty(M)$, is a Lie algebra, and it is the Lie algebra of $\operatorname{Diff}(M)$.

Proof. The bracket of derivations is a derivation, so it is a vector field; it is bilinear, alternating, and satisfies the Jacobi identity because it is the commutator in the associative algebra of derivations. The identification with $T_{\mathrm{id}}\operatorname{Diff}(M)$ is the standard identification of a vector field with an infinitesimal flow.

Proposition (failure of the local diffeomorphism property). The exponential map of $\operatorname{Diff}(M)$ is singular at every nonzero vector field whose flow has a period, and it is not locally surjective: its image contains no neighbourhood of the identity.

Proof sketch. A nonzero vector field $X$ with a periodic flow of period $1$, for instance a constant vector field on the circle $S^1$, satisfies $\exp(X) = \mathrm{id}$; varying $X$ in the direction of the period leaves the time-one map trivial, so $d(\exp)_X$ has a nontrivial kernel. The failure of local surjectivity is established by exhibiting diffeomorphisms arbitrarily close to the identity that are not the time-one map of any flow; the argument counts the dimension available in the algebra against the degrees of freedom of the group near the identity.

Example. On $M = S^1$, the vector field $X = \partial_\theta$ has flow $\Phi_X^t(\theta) = \theta + t$, so $\exp(2\pi k\, \partial_\theta) = \mathrm{id}$ for every integer $k$; the exponential is far from injective, and the Lie algebra $\mathrm{X}(S^1)$ is infinite-dimensional while the group is a union of finite-dimensional circles.

Summary

Hilbert's fifth problem asks whether a topological group that is a locally Euclidean topological manifold is necessarily a Lie group, and the answer is affirmative. A locally Euclidean group is locally compact and has no small subgroups; by the theorem of Gleason and Montgomery–Zippin, a locally compact group with no small subgroups is a Lie group, with a unique smooth structure. For a general locally compact group, Yamabe's theorem gives an open subgroup that is a projective limit of Lie groups, and equivalently a compact normal subgroup whose quotient is a Lie group; this describes the boundary of the problem and shows where the locally Euclidean hypothesis is used.

In infinite dimensions the model space replaces local compactness. For Banach–Lie groups the exponential map is a local diffeomorphism, the locally convex structure supports an inverse function theorem, the Campbell–Baker–Hausdorff series converges and the Lie correspondence holds. For Fréchet–Lie groups these statements fail: the exponential map of the diffeomorphism group of a compact manifold is singular at the vector fields with periodic flow and is not locally surjective, and the BCH series need not converge. Milnor's notion of a regular Lie group isolates the cases in which the BCH product is defined and smooth and the group law is recovered from the algebra; the diffeomorphism group of a compact manifold is regular, and the Nash–Moser inverse function theorem is the analytic tool that supplies the smoothness statements of the theory.

Summary of Notation

Symbol	Meaning
$\mathbb{K}$	$\mathbb{R}$ or $\mathbb{C}$; the scalar field of the Lie groups and algebras below
$G$	Topological group; Lie group when smooth
$\mathrm{G} = T_eG$	Lie algebra of a Lie group
NSS	No small subgroups: some neighbourhood of $e$ contains no nontrivial subgroup
Gleason–Montgomery–Zippin	Locally compact $+$ NSS $\Rightarrow$ Lie group
Yamabe	Locally compact $G$ has an open subgroup a projective limit of Lie groups; $G_0/K$ a Lie group with $K$ compact normal
Banach–Lie group	Lie group modelled on a Banach space; $\exp$ a local diffeomorphism
Fréchet–Lie group	Lie group modelled on a Fréchet space; $\exp$ need not be a local diffeomorphism
$\mathrm{X}(M)$	Lie algebra of smooth vector fields on $M$; bracket $[X,Y] = X\circ Y - Y\circ X$
$\operatorname{Diff}(M)$	Fréchet–Lie group of diffeomorphisms of a compact manifold $M$
$\exp(X) = \Phi_X^1$	Exponential of a vector field: time-one map of its flow
Regular Lie group	The BCH product is defined and smooth; the group law is determined by the algebra
Tame Fréchet space, tame map	Fréchet space and map with quantified loss-of-derivative estimates
Nash–Moser	Inverse function theorem for tame maps; the Fréchet replacement for the classical theorem
$\varprojlim G_i$	Projective limit of an inverse system of Lie groups
$Z(X,Y)$, $\exp X\exp Y = \exp Z(X,Y)$	Campbell–Baker–Hausdorff series
Totally disconnected	The only connected subset containing $e$ is $\{e\}$; van Dantzig gives a compact open subgroup
$\mathbb{Z}_p$	$p$-adic integers; compact, totally disconnected, not a Lie group
Hilbert–Smith	A locally compact group acting faithfully on a connected manifold is a Lie group