Convenient vector space

The c∞-topology

Let E be a locally convex vector space. A curve c : ℝ → E is called smooth or C∞
if all derivatives exist and are continuous. Let C∞
(ℝ,E) be the space of smooth curves. It can be shown that the set of smooth curves does not depend entirely on the locally convex topology of E, only on its associated bornology (system of bounded sets); see [KM], 2.11. The final topologies with respect to the following sets of mappings into E coincide; see [KM], 2.13.

This topology is called the c∞
-topology on E and we write c∞
E for the resulting topological space. In general (on the space D of smooth functions with compact support on the real line, for example) it is finer than the given locally convex topology, it is not a vector space topology, since addition is no longer jointly continuous. Namely, even c∞
(D × D) ≠ (c∞
D) × (c∞
D). The finest among all locally convex topologies on E which are coarser than c∞
E is the bornologification of the given locally convex topology. If E is a Fréchet space, then c∞
E = E.

Convenient vector spaces

A locally convex vector space E is said to be a convenient vector space if one of the following equivalent conditions holds (called c∞
-completeness); see [KM], 2.14.

Here a mapping f : ℝ → E is called Lipk
if all derivatives up to order k exist and are Lipschitz, locally on R.

Smooth mappings

Let E and F be convenient vector spaces, and let U ⊆ E be c∞
-open. A mapping f : U → F is called smooth or C∞
, if the composition f o c ∈ C∞
(ℝ,F) for all c ∈ C∞
(ℝ,U). See[KM], 3.11.

Main properties of smooth calculus

1. For maps on Fréchet spaces this notion of smoothness coincides with all other reasonable definitions. On ℝ2
this is a non-trivial theorem, proved by Boman, 1967. See also [KM], 3.4.

2. Multilinear mappings are smooth if and only if they are bounded ([KM], 5.5).

3. If f: E ⊇ U → F is smooth then the derivative df : U × E → F is smooth, and also df : U → L(E,F) is smooth where L(E,F) denotes the space of all bounded linear mappings with the topology of uniform convergence on bounded subsets; see [KM], 3.18.

4. The chain rule holds ([KM], 3.18).

5. The space C∞
(U,F) of all smooth mappings U → F is again a convenient vector space where the structure is given by the following injection, where C∞
(ℝ,ℝ) carries the topology of compact convergence in each derivative separately; see [KM], 3.11 and 3.7.

6. The exponential law holds ([KM], 3.12): For c∞
-open V ⊆ F the following mapping is a linear diffeomorphism of convenient vector spaces.

This is the main assumption of variational calculus. Here it is a theorem. This property is the source of the name convenient, which was borrowed from (Steenrod 1967).

7. Smooth uniform boundedness theorem ([KM], theorem 5.26). A linear mapping f : E → C∞
(V,G) is smooth (by (2) equivalent to bounded) if and only if ev
v o f : V → G is smooth for each v ∈ V.

8. The following canonical mappings are smooth. This follows from the exponential law by simple categorical reasonings, see [KM], 3.13.

Related convenient calculi

Convenient calculus of smooth mappings appeared for the first time in [Frölicher, 1981], [Kriegl 1982, 1983]. Convenient calculus (having properties 6 and 7) exists also for:

The corresponding notion of convenient vector space is the same (for their underlying real vector space in the complex case) for all these theories.

Application: Manifolds of mappings between finite dimensional manifolds

The exponential law 6 of convenient calculus allows for very simple proofs of the basic facts about manifolds of mappings. Let M and N be finite dimensional smooth manifolds where M is compact. We use an auxiliary Riemann metric g ¯ on N. The Riemannian exponential mapping of g ¯ is described in the following diagram:

It induces an atlas of charts on the space C∞
(M , N) of all smooth mappings M → N as follows. A chart centered at f ∈ C∞
(M,N), is:

Now the basics facts follow in easily. Trivializing the pull back vector bundle f∗
TN and applying the exponential law 6 leads to the diffeomorphism

All chart change mappings are smooth (C∞
) since they map smooth curves to smooth curves:

Thus C∞
(M , N) is a smooth manifold modelled on Fréchet spaces. The space of all smooth curves in this manifold is given by

Since it visibly maps smooth curves to smooth curves, composition

is smooth. As a consequence of the chart structure, the tangent bundle of the manifold of mappings is given by

Regular Lie groups

Let G be a connected smooth Lie group modeled on convenient vector spaces, with Lie algebra g = T e G . Multiplication and inversion are denoted by:

The notion of a regular Lie group is originally due to Omori et al. for Fréchet Lie groups, was weakened and made more transparent by J.Milnor, and was then carried over to convenient Lie groups; see [KM], 38.4.

A Lie group G is called regular if the following two conditions hold:

If g is the unique solution for the curve X required above, we denote

If X is a constant curve in the Lie algebra, then {{{1}}} is the group exponential mapping.

Theorem. For each compact manifold M, the diffeomorphism group Diff(M) is a regular Lie group. Its Lie algebra is the space X ( M ) of all smooth vector fields on M, with the negative of the usual bracket as Lie bracket.

Proof: The diffeomorphism group Diff(M) is a smooth manifold since it is an open subset in C∞
(M , M). Composition is smooth by restriction. Inversion is smooth: If t → f(t,  ) is a smooth curve in Diff(M), then f(t,  )−1
satisfies the implicit equation f ( t , f ( t , ) − 1 ( x ) ) = x , so by the finite dimensional implicit function theorem, (t.x) ↦ f(t,  )−1
(x) is smooth. So inversion maps smooth curves to smooth curves, and thus inversion is smooth. Let X(t,x) be a time dependent vector field on M (in C ∞ ( R , X ( M ) ) ). Then the flow operator Fl of the corresponding autonomous vector field ∂ t × X on ℝ × M induces the evolution operator via

which satisfies the ordinary differential equation

Given a smooth curve in the Lie algebra, X ( s , t , x ) ∈ C ∞ ( R 2 , X ( M ) ) , then the solution of the ordinary differential equation depends smoothly also on the further variable s, thus evolr
Diff(M maps smooth curves of time dependent vector fields to smooth curves of diffeomorphism. QED.

The principal bundle of embeddings

For finite dimensional manifolds M and N with M compact, the space Emb(M,N) of all smooth embeddings of M into N, is open in C∞
(M , N), so it is a smooth manifold. The diffeomorphism group Diff(M) acts freely and smoothly from the right on Emb(M,N).

Theorem: Emb(M,N) → Emb(M,N)/Diff(M) is a principal fiber bundle with structure group Diff(M).

Proof: One uses again an auxiliary Riemannian metric g ¯ on N. Given f ∈ Emb(M,N), view f(M) as a submanifold of N, and split the restriction of the tangent bundle TN to f(M) into the subbundle normal to f(M) and tangential to f(M) as T N | f ( M ) = Nor ⁡ ( f ( M ) ) ⊕ T f ( M ) . Choose a tubular neighborhood

If g: M → N is C1
-near to f, then

This is the required local splitting. QED

Further applications

An overview of applications using geometry of shape spaces and diffeomorphism groups can be found in [Bauer, Bruveris, Michor, 2014].